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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