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### Second-Order Linear Inhomogeneous Equations

Solve the initial value problem \begin{align} \frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 5y &= \cos(x) \\ y(0) &= -9 \\ 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. $$(D^2 - 4D + 5)y = 0$$ Substep 2: Find the roots. We use the quadratic formula to compute the roots, $2\pm i$

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

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

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 $(4A - 4B)\cos(x) + (4A + 4B)\sin(x) = \cos(x).$ This gives us the equations $4A - 4B=1$ and $4A + 4B=0.$ From these equations we get $A=1/8$ and $B=-1/8.$ So we get $$y_p(x)=(1/8)\cos(x) - (1/8)\sin(x).$$ 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 $$y(x) = (1/8)\cos(x) - (1/8)\sin(x) + c_1\exp(2x)\cos(x) + c_2\exp(2x)\sin(x)$$ 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, $y'(x) = -(1/8)\cos(x) - (1/8)\sin(x) + 2c_1\exp(2x)\cos(x) - c_1\exp(2x)\sin(x) + 2c_2\exp(2x)\sin(x) + c_2\exp(2x)\cos(x).$ Then we plug in $x=0$ to get the following equations. \begin{align} y(0) = 1/8 + c_1 &= -9 \\y'(0) = -1/8 + 2c_1 + c_2 &= -5 \end{align} We solve these equations to get $c_1=-73/8$ and $c_2=107/8.$ Finally, we plug our values for $c_1$ and $c_2$ into the general solution to find our solution to the initial value problem. $$y(x) = (1/8)\cos(x) - (1/8)\sin(x) - (73/8)\exp(2x)\cos(x) + (107/8)\exp(2x)\sin(x).$$ You may reload this page to generate additional examples.