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Second-Order Linear Inhomogeneous Equations

Additional Examples

Solve the initial value problem $$ \begin{align} \frac{d^2y}{dx^2} - 10\frac{dy}{dx} + 9y &= x - 2 \\ y(0) &= -3 \\ y'(0) &= 4 \end{align} $$

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. $$ (D^2 - 10D + 9)y = 0 $$ Substep 2: Find the roots. The equation factors into $ (D - 1)(D - 9)y = 0.$ So our roots are 1 and 9.

Substep 3: Write the homogeneous solutions. $$ y_h(x) = c_1\exp(x) + c_2\exp(9x) $$ Step 2: Guess the form of the particular solutions

Since the right-hand side is $x - 2,$ we guess $ y_p(x) = Ax + B.$

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 $ 9Ax + 9B - 10A = x - 2.$ This gives us the equations $9A=1$ and $9B - 10A=-2.$ From these equations we get $A=1/9$ and $B=-8/81.$ So we get $$ y_p(x)=(1/9)x - 8/81. $$ 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 $$ y(x) = (1/9)x - 8/81 + c_1\exp(x) + c_2\exp(9x) $$ 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) = 1/9 + c_1\exp(x) + 9c_2\exp(9x).$ Then we plug in $ x=0$ to get the following equations. $$ \begin{align} y(0) = -8/81 + c_1 + c_2 &= -3 \\y'(0) = 1/9 + c_1 + 9c_2 &= 4 \end{align} $$ We solve these equations to get $ c_1=-15/4$ and $ c_2=275/324.$ Finally, we plug our values for $ c_1$ and $ c_2$ into the general solution to find our solution to the initial value problem. $$ y(x) = (1/9)x - 8/81 - (15/4)\exp(x) + (275/324)\exp(9x). $$ You may reload this page to generate additional examples.


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©2010, 2014 Andrew G. Bennett