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Second-Order Linear Inhomogeneous Equations

Additional Examples

Solve the initial value problem $$ \begin{align} \frac{d^2y}{dx^2} + 2\frac{dy}{dx} - 3y &= -3\sin(2x) \\ y(0) &= -7 \\ y'(0) &= 5 \end{align} $$

This is a second-order linear constant-coefficient initial value problem.

First, we find the general solution

Step 1: Find the homogeneous solution.

Substep 1: Write the equation in operator form. $$ (D^2 + 2D - 3)y = 0 $$ Substep 2: Find the roots. The equation factors into $ (D + 3)(D - 1)y = 0.$ So our roots are -3 and 1.

Substep 3: Write the homogeneous solutions. $$ y_h(x) = c_1\exp(-3x) + c_2\exp(x) $$ Step 2: Guess the form of the particular solutions

Since the right-hand side is $-3\sin(2x),$ we guess $ y_p(x) = A\cos(2x) + B\sin(2x).$

Step 3: Plug our guess into the equation and solve for the undetermined coefficients

Plugging our guess for $ y_p(x)$ into the equation, we obtain $ (-7A + 4B)\cos(2x) + (-4A - 7B)\sin(2x) = -3\sin(2x).$ This gives us the equations $-7A + 4B=0$ and $-4A - 7B=-3.$ From these equations we get $A=12/65$ and $B=21/65.$ So we get $$ y_p(x)=(12/65)\cos(2x) + (21/65)\sin(2x). $$ Step 4: The general solution is the particular solution plus all the homogeneous solutions.

So from the results of steps 1 and 3, we get the general solution is $$ y(x) = (12/65)\cos(2x) + (21/65)\sin(2x) + c_1\exp(-3x) + c_2\exp(x) $$ Second, we solve for the constants

Now that we have the general solution, we plug in our initial values to find the solution to the initial value problem. First we compute, $ y'(x) = (42/65)\cos(2x) - (24/65)\sin(2x) - 3c_1\exp(-3x) + c_2\exp(x).$ Then we plug in $ x=0$ to get the following equations. $$ \begin{align} y(0) = 12/65 + c_1 + c_2 &= -7 \\y'(0) = 42/65 - 3c_1 + c_2 &= 5 \end{align} $$ We solve these equations to get $ c_1=-75/26$ and $ c_2=-43/10.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem. $$ y(x) = (12/65)\cos(2x) + (21/65)\sin(2x) - (75/26)\exp(-3x) - (43/10)\exp(x). $$ You may reload this page to generate additional examples.


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