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Second-Order Linear Inhomogeneous Equations

Additional Examples

Solve the initial value problem $$ \begin{align} \frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y &= 3\cos(3x) \\ y(0) &= 5 \\ y'(0) &= -4 \end{align} $$

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

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

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

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 - 9B)\cos(3x) + (9A - 7B)\sin(3x) = 3\cos(3x).$ This gives us the equations $-7A - 9B=3$ and $9A - 7B=0.$ From these equations we get $ A=-21/130$ and $B=-27/130.$ So we get $$ y_p(x)=-(21/130)\cos(3x) - (27/130)\sin(3x). $$ 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 $$ y(x) = -(21/130)\cos(3x) - (27/130)\sin(3x) + c_1\exp(x) + c_2\exp(2x) $$ 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) = -(81/130)\cos(3x) + (63/130)\sin(3x) + c_1\exp(x) + 2c_2\exp(2x).$ Then we plug in $ x=0$ to get the following equations. $$ \begin{align} y(0) = -21/130 + c_1 + c_2 &= 5 \\y'(0) = -81/130 + c_1 + 2c_2 &= -4 \end{align} $$ We solve these equations to get $ c_1=137/10$ and $ c_2=-111/13.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem. $$ y(x) = -(21/130)\cos(3x) - (27/130)\sin(3x) + (137/10)exp(x) - (111/13)\exp(2x). $$ You may reload this page to generate additional examples.


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