Kansas State University

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. $$ (D^2 + 16)y = 0 $$ Substep 2: Find the roots. We use the quadratic formula to compute the roots, $ \pm 4i$

Substep 3: Write the homogeneous solutions. $$ y_h(x) = c_1\cos(4x) + c_2\sin(4x) $$ Step 2: Guess the form of the particular solutions

Since the right-hand side is $-5\sin(2x),$ 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 $ 12A\cos(2x) + 12B\sin(2x) = -5\sin(2x).$ This gives us the equations $12A=0$ and $12B=-5.$ From these equations we get $A=0$ and $B=-5/12.$ So we get $$ y_p(x)=-(5/12)\sin(2x). $$ 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 $$ y(x) = -(5/12)\sin(2x) + c_1\cos(4x) + c_2\sin(4x) $$ 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) = -(5/6)\cos(2x) - 4c_1sin(4x) + 4c_2\cos(4x).$ Then we plug in $ x=0$ to get the following equations. $$ \begin{align} y(0) = c_1 &= -9 \\y'(0) = -5/6 + 4c_2 &= 7 \end{align} $$ We solve these equations to get $ c_1=-9$ and $ c_2=47/24.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem. $$ y(x) = -(5/12)\sin(2x) - 9\cos(4x) + (47/24)\sin(4x). $$ You may reload this page to generate additional examples.