MATH 417 Lecture 17

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Review

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} y'(t) = f(t,y(t)) \\ y(a) = \alpha \end{cases}}

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \in \left[a,b\right]} .

Numerical Methods:

  1. Euler's Method:
    • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_0 = \alpha}
    • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = w_n + h \, f(t_h, w_h)}
    • error is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| y(t_n) - w_n \right| \le C \, h \in O(h)}
    • one-step error is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underbrace{y_{n+1}}_{\mbox{exact}} - \left( y_n + h\, f(t_n,y_n) \right) \in O(h^2)} (but these errors accumulate).
  2. Taylor Series-based Method:
    • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \, T^{(n)}(t,y) = h\, \left( f(t,y) + \frac{h}{2} \cdot \frac{\mathrm{d}f(t,y(t)}{\mathrm{d}t} + \frac{h^2}{3!} \cdot \frac{\mathrm{d}^2 f(t,y(t))}{\mathrm{d}t^2} + \dots \right)}
    • Need to know partial derivatives
    • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{(n)}} local error = , and global error is
    • Euler's method is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{(1)}} .

Runge-Kutta Method

Replaces derivatives with nested function evaluations.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = w_n + h \, f(t_n + \alpha, w_n + \beta f(t_n, w_n)) = }

We want Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = \dots = y_{n+1} + O(h^3)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t+h) = y(t) + h \, y'(t) + \frac{h^2}{2} \, y''(t) + O(h^3)}

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{(2)}} method uses Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t+h) \approx y(t) + h \, f(t,y(t)) + \frac{h^2}{2} \, \left( f_t(t,y(t)) + f_y(t,y(t)) \, f(t,y(t)) \right) + O(h^3)}


Second order Runge-Kutta method (RK-2) uses Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t+h) \approx y + h \, f(t + \alpha, y + \beta \, f(t, y)) + O(h^3)}


Hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t) + h \, f(t,y(t)) + \frac{h^2}{2} \, \left( f_t(t,y(t)) + f_y(t,y(t)) \, f(t,y(t)) \right) = y(t+h) + h \, f(t + \alpha, y(t) + \beta \, f(t, y(t)))} up to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(h^3)} difference.

Taylor Series for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t,y)} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t + \Delta t, y + \Delta y) = f(t,y) + \Delta t \, f_t(t,y) + \Delta y \, f_y(t,y) + \frac{1}{2!} \, \left( \Delta t^2 \, f_{tt}(t,y) + 2\Delta t \, \Delta y \, f_{ty}(t,y) + \Delta y^2 \, f_{yy}(t,y) \right) + \dots + \frac{1}{n!} \, \left( \sum_{k=0}^n \left( \Delta t \right)^{h-k} \, \left( \Delta y \right)^k \, \binom{n}{k} \, \frac{\partial^{n-k}}{\partial t^{n-k}} \, \frac{\partial^k}{\partial y^k} \, f(t,y) \right) + O \left( \left| \Delta t \right|^{n+1} + \left| \Delta y \right|^{n+1} \right)}


RK-2: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t+h) \approx y + h\,f(t-\alpha, y + \beta \, f(t,y))} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha,\beta \in O(h)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} y(t+h) &\approx y + h \, f(t-\alpha, y + \beta \, f(t,y)) \\ &= y(t) + h \, \left[ f(t,y) + \alpha \, f_t(t,y) + \beta \, f(t,y) \cdot f_y(t,y) + O(\alpha^2 + (\beta \, f)^2 \right] \\ &= y(t) + h \, f(t,y(t)) + \alpha \, h \, f_t + \beta \, h \, f \, f_y + h \, O(h^2 + h^2 \, f^2) \\ &= T^{(2)}(t,y) \quad \iff \quad \alpha = \frac{h}{2}; \beta = \frac{h}{2} \end{align}}

Hence RK-2 is the midpoint method: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = w_n + h \, f \left( t_n + \frac{h}{2}, w_n + \frac{h}{2} \, f(t_n, w_n) \right)}

If we let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1 = h \, f(t_n, w_n)} , we can write Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = w_n + h \, f(t_n + \frac{h}{2}, w_n + \frac{1}{2} \, k_1}

Here comes the rule:

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_0 = \alpha}
  2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 0, 1, 2, \ldots}
  3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1 = h \, f(t_n, w_n)}
  4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = w_n + h \, f \left( t + \frac{h}{2} , w_n + \frac{1}{2} k_1 \right)}


Modified Euler

Based on trapezoidal rule to approximate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t_n + h) = y(t_n) + \int_{t_n}^{t_n + h} f(t_n, y(t_n)) \,\mathrm{d}t}


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} y(t_n + h) &\approx y(t_n) + \frac{h}{2} \, \left( y'(t_n) + y'(t_{n+1}) \right) \\ &\approx y(t_n) + \frac{h}{2} \, \left( f(t_n, y(t_n)) + f(t_n + h, y(t_n + h)) \right) \end{align}}

Use Euler's method to find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t_n + h) \approx y(t_n) + h \, f(t_n, y(t_n))}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} y(t_n + h) &\approx y(t_n) + \frac{h}{2} \, \left( f(t_n, y(t_n)) + f(t_n + h, y(t_n) + h \, f(t_n, y(t_n))) \right) \end{align}}

To find our numeric method, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_n = y(t_n)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = y(t_n + h)} , then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} w_{n+1} &= w_n + \frac{h}{2} \, \left( f(t_n, w_n) + f(t_{n+1}, w_n + h \, f(t_n, w_n)) \right) \\ \end{align}}


Heun's Method

Based on gaussian quadrature method to approximate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{n+1} = y_n + \int_{t_n}^{t_n + h} f(t, y(t) \,\mathrm{d}t \approx w_n + \frac{h}{4} \, f(t_n, w_n) + \frac{3h}{4} \, f \left( t_n + \frac{2h}{3}, y \left( t_n + \frac{2h}{3} \right) \right)}

Here's the rule:

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_0 = \alpha}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 0, 1, 2, \ldots}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = w_n + \frac{h}{4} \, f(t_n, w_n) + \frac{3h}{4} \, f \left( t_n + \frac{2h}{3}, w_n + \frac{2h}{3} \, f(t_n, w_n) \right)}

The book has an error by saying this is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(h^3)} . It is RK-2, so it must be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(h^2)} .


RK-4

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_0 = \alpha}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 0, 1, 2, \ldots}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1 = h \, f(t_n, w_n)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_2 = h \, f \left( t_n + \frac{h}{2}, w_n + \frac{1}{2} \, k_1 \right)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_3 = h \, f \left( t_n + \frac{h}{2}, w_n + \frac{1}{2} \, k_2 \right)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_4 = h \, f \left( t_n + h, w_n + k_3 \right)}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = w_n + \frac{1}{6} \, \left( k_1 + 2k_2 + 2k_3 + k_4 \right)}


Multi-Step Method

All methods we've discussed so far are 'one step methods: we only use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_n} to approximate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1}} .


Assume that you have computed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ldots, (t_{n-1}, w_{n-1}); (t_n,w_n)} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = w_n + \frac{h}{720} \, \left( 1901 \, f(t_n, w_n) - 2774 \, f(t_{n-1}, w_{n-1}) + 2616 \, f(t_{n-2}, w_{n-2}) - 1274 \, f(t_{n-3}, w_{n-3}) + 251 \, f(t_{n-4}, w_{n-4}) \right)}

This is an order 5 method, but it has oscillating coefficients (so it is numerically unstable)


Examples

No Solution Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} y'(t) = (y(t))^2 \\ y(0) = 1 \end{cases}}

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t,y) = y^2} , which is continuous
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \frac{\partial f}{\partial y} \right| = \left| 2y \right| \not < C}

Exact solution is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t) = \frac{1}{1-t}} , the reason the lipschitz discontinuity is because there is a singularity at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = 1}

Example 2

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} y'(t) = y(t) + t \\ y(0) = 1 \end{cases}}

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t,y) = y + t} (continuous)
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \frac{\partial f}{\partial y} \right| = 1} , so Lipschitz continuous

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le t \le 1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h = 0.5} , let's find Euler, Midpoint, and Modified Euler approximations:


Euler

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_0 = 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1 = \frac{1}{2}} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_2 = 1} .

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_0 = 1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_1 = w_0 + h \, f(t_0, w_0) = 1 + \frac{1}{2} \, f(0,1) = \frac{3}{2}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_2 = w_1 + h \, f(t_1, w_1) = \frac{3}{2} + \frac{1}{2} \, f \left( \frac{1}{2} , \frac{3}{2} \right) = \frac{5}{2}}


Midpoint

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = w_n + \frac{h}{2} \, f \left( t_n + \frac{h}{2}, w_n + \frac{h}{2} \, f(t_n, w_n) \right)}

Something's wrong here...
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_0 = 1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_1 = 1 + \frac{1}{2} \, f \left( \frac{1}{4}, 1 + \frac{1}{4} \, f(0,1) \right) = 1 + \frac{3}{8} = \frac{11}{8}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_2 = \frac{11}{8} + \frac{1}{2} \, f \left( \frac{3}{4} + \frac{11}{8}, 1 + \frac{1}{4} \, f \left( \frac{1}{2}, \frac{11}{8} \right) \right) = \ldots}


Modified Euler

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n+1} = w_n + \frac{h}{2} \, \left( f(t_n, w_n) + f(t_n + h, w_n + h \, f(t_n, w_n)) \right)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1 = h \, f(t_n, w_n)} simplifies the above to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_n + \frac{1}{2} \, k_1 + \frac{h}{2} \, f(t_n + k_1, w_n + k_1)}

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_0 = 1}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_1 = 1 + \frac{1}{4} \, \left( f(0,1) + f \left( \frac{1}{2}, 1 + \frac{1}{2} \, f(0,1) \right) \right) = \frac{7}{4}}
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{7}{4} + \frac{1}{4} \, \left( f \left( \frac{1}{2}, \frac{7}{4} \right) + f \left( 1 , \frac{7}{4} + \frac{1}{2} \, f \left( \frac{1}{2}, \frac{7}{4} \right) \right) \right) = \ldots}
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