MATH 308 Lecture 15

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Lecture Notes


Remark for section 3.5

How would we go about solving ?

Particular Solution


Section 3.6: Variation of Parameters

Find solution to

Homogeneous solution is

What about non-constant coefficients; i.e. what if the constants were actually functions of :

There are infinitely many possibilities for and , so we impose a restriction to limit the number of possibilities:


Using the above examples, we let and :

Simplification: We were able to simplify with the restriction .

Now we can plug our functions into the differential equation:

Use restriction as second equation in system:

We solve for and :

So general solution to original differential equation is:

Method for Solving Second Order ODEs

To determine a particular solution to

Find and for corresponding 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'' + p(x) \, y' + q(x) \, y = 0}

Solve the following system for 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(x)}

or

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} c_1(x) &= - \int_{x_0}^x \frac{y_2(s) \, g(s)}{W(y_1,y_2)(s)} \, \mathrm{d}s & c_2(x) &= \int_{x_0}^x \frac{y_1(s) \, g(s)}{W(y_1,y_2)(s)} \, \mathrm{d}s \end{align}}

A particular solution will be


Exercise 7

Find general solution 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 y'' + y = 0}

Solution to homogeneous equation is , 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 y_1 = \cos{x}} 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 = \sin{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 W(y_1,y_2) = \cos^2(x) + \sin^2(x) = 1}

Plug in to equations for 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}

General solution is