MATH 308 Lecture 10

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Lecture Notes

Chapter 3

Solving second order linear differential equations

Section 3.1–3.4: Homogeneous with constant coefficients

P(x)y''+Q(x)y'+R(x)y=G(x)

When $G(x)=0$ , equation is homogeneous When $P(x)$ , $Q(x)$ , and $R(x)$ do not depend on 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 x} , the equation has constant coefficients

Examples:

Equation	Linear	Homogeneous	Constant Coefficients
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 x\,y'' + 3y' + y\,\sin{x} = \mathrm{e}^x}	Yes	No	No
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'' + x\,y' = (y-x)x}	Yes	No	No
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'' + x\,y\,y' = x \, y}	No	--	--
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'' = y\,\sin{x}}	Yes	Yes	No
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'' + 5y' + 3y = \sin{x}}	Yes	No	Yes
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'' - 2y' = 5 \sqrt{y}}	No	--	--

Theorem

Let $y_{1}$ 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 y_2} be two solutions to a second order homogeneous linear differential equation. Any linear combination 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 c_1 \, y_1 + c_2 \, y_2} , for any real 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 c_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 c_2} is a solution to the differential equation.

Exercise 2

Given that 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_1(x) = \mathrm{e}^{3x}} and $y_{2}(x)=\mathrm {e} ^{2x}$ are solutions to the homogeneous equation

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'' - 5y' + 6y = 0}

Show that every linear combination 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 c_1 \, y_1 + c_2 \, y_2} for any 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 c_1, c_2 \in \mathbb{Z}} is a solution to the homogeneous problem.

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} f(x) &= c_1 \, \mathrm{e}^{3x} + c_2 \, \mathrm{e}^{2x} f'(x) &= c_1 \, \mathrm{e}^{3x} + c_2 \, \mathrm{e}^{2x} \end{align}}

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} f(x) &= c_1 \, y_1 + c_2 \, y_2 \\ f'(x) &= c_1 \, y_1' + c_2 \, y_2' \\ f''(x) &= c_1 \, y_1'' + c_2 \, y_2'' \\ \end{align}}

${\begin{aligned}c_{1}\,y_{1}''+c_{2}\,y_{2}''-5(c_{1}\,y_{1}'+c_{2}\,y_{2}')+6(c_{1}\,y_{1}+c_{2}\,y_{2})&=0\\c_{1}(y_{1}''-5y_{1}'+6y_{1})+c_{2}(y_{2}''-5y_{2}'+6y_{2})&=0\\\end{aligned}}$

y_1 is a solution: 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_1'' - 5y_1' + 6y_1 = 0}
y_2 is a solution: 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_2'' - 5y_2' + 6y_2 = 0}

Therefore 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} is a solution.

Find a solution that satisfies the initial condition 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(0) = 3} 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 y'(0) = -1} .

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 f} in the form $c_{1}\,\mathrm {e} ^{3x}+c_{2}\,\mathrm {e} ^{2x}$

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} f(x) &= c_1 \, \mathrm{e}^{3x} + c_2 \, \mathrm{e}^{2x} \\ f'(x) &= 3 c_1 \, \mathrm{e}^{3x} + 2 c_2 \, \mathrm{e}^{2x} \\ f(0) &= c_1 + c_2 = 3 \\ f'(0) &= 3 c_1 + 2 c_2 = -1 \\ c_1 &= -7 \\ c_2 &= 10 \end{align}}

Therefore, the 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 -7\mathrm{e}^{3x} + 10\mathrm{e}^{2x}}

Exercise 3

Find the general solution to the differential equation 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'' - 7y' + 12 y = 0} . Look for a solution of the form 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 = \mathrm{e}^{r\,x}} .

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 &= \mathrm{e}^{rx} \\ y' &= r \, \mathrm{e}^{rx} \\ y'' &= r^2 \, \mathrm{e}^{rx} \end{align}}

So we have

${\begin{aligned}r^{2}\,\mathrm {e} ^{rx}-7r\,\mathrm {e} ^{rx}+12\mathrm {e} ^{rx}&=0\\r^{2}-7r+12&=0\end{aligned}}$

This is called the characteristic equation.

Solving 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 r} gives 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 \{3,4\}} , so the following formulae are solutions:

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_1 = \mathrm{e}^{3x}}
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_2 = \mathrm{e}^{4x}}

And any linear combination 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 = c_1 \, \mathrm{e}^{3x} + c_2 \mathrm{e}^{4x}} is the general solution to the differential equation.

Characteristic Equation

Replace each derivative with a coefficient of the same degree.

For 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 y'' - 3y' - 10y = 0} becomes 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 r^2 - 3r - 10 = 0} .

Solving the characteristic gives coefficients for the exponential.

Exercise 4

Find the general solutino to the differential solution 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'' + 4y = 0} .

Characteristic equation 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 r^2 + 4 = 0} , and roots are $r=\pm 2i$ .

General complex valued 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 = c_1 \, \mathrm{e}^{2i\,x} + c_2 \, \mathrm{e}^{-2i\,x}} .

Recall the Euler Formula: 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 \mathrm{e}^{i \,\theta} = \cos{\theta} + i \, \sin{\theta}}

The solution takes the form

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_1 &= \cos{2x} + i \, \sin{2x} \\ y_2 &= \cos{-2x} + i \, \sin{-2x} = \cos{2x} - i \, \sin{2x} \\ \end{align}}

Recombining we get

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 c_1 = c_2 = \frac{1}{2}} , 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 \frac{y_1 + y_2}{2} = \cos{2x}}
Let $c_{1}=c_{2}={\frac {1}{2i}}$ , 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 \frac{y_1 + y_2}{2} = \sin{2x}} .

So we can express the solution as a linear combination of these two real functions:

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 = c_1 \, \cos{2x} + c_2 \, \sin{2x}}