7 Inhomogeneous term in the complementary function

Occasionally you will come across a differential equation $a\frac{d^{2}y}{d{x}^2}+b\frac{dy}{dx}+cy=f\left(x\right)$ for which the inhomogeneous term, $f\left(x\right)$ , forms part of the complementary function. One such example is the equation

$\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}-6y={\text{e}}^{3x}$

It is straightforward to check that the complementary function is $y_{\text{cf}}=A{\text{e}}^{3x}+B{\text{e}}^{-2x}.$ Note that the first of these terms has the same form as the inhomogeneous term, ${\text{e}}^{3x}$ , on the right-hand side of the differential equation.

You should verify for yourself that trying a particular integral of the form $y_{\text{p}}\left(x\right)=\alpha {\text{e}}^{3x}$ will not work in a case like this. Can you see why?

Instead, try a particular integral of the form $y_{\text{p}}\left(x\right)=\alpha x{\text{e}}^{3x}$ . Verify that

$\frac{d{y}_{\text{p}}}{dx}=\alpha {\text{e}}^{3x}\left(3x+1\right)\text{and}\frac{d^2{y}_{\text{p}}}{d{x}^2}=\alpha {\text{e}}^{3x}\left(9x+6\right).$

Substitute these expressions into the differential equation to find $\alpha =\frac{1}{5}$ .

Finally, the particular integral is $y_{\text{p}}\left(x\right)=\frac{1}{5}x{\text{e}}^{3x}$ and so the general solution to the differential equation is:

$y=A{\text{e}}^{3x}+B{\text{e}}^{-2x}+\frac{1}{5}x{\text{e}}^{3x}$

This shows a generally effective method - where the inhomogeneous term $f\left(x\right)$ appears in the complementary function use as a trial particular integral $x$ times what would otherwise be used.

Exercises

1. Find the general solution of the following equations:
   1. $\frac{d^{2}x}{d{t}^2}-2\frac{dx}{dt}-3x=6$  
   2. $\frac{d^{2}y}{d{x}^2}+5\frac{dy}{dx}+4y=8$
   3. $\frac{d^{2}y}{d{t}^2}+5\frac{dy}{dt}+6y=2t$
   4. $\frac{d^{2}x}{d{t}^2}+11\frac{dx}{dt}+30x=8t$
   5. $\frac{d^{2}y}{d{x}^2}+2\frac{dy}{dx}+3y=2sin2x$
   6. $\frac{d^{2}y}{d{t}^2}+\frac{dy}{dt}+y=4cos3t$
   7. $\frac{d^{2}y}{d{x}^2}+9y=4{\text{e}}^{8x}$    
   8. $\frac{d^{2}x}{d{t}^2}-16x=9{\text{e}}^{6t}$
2. Find a particular integral for the equation $\frac{d^{2}x}{d{t}^2}-3\frac{dx}{dt}+2x=5{\text{e}}^{3t}$
3. Find a particular integral for the equation $\frac{d^{2}x}{d{t}^2}-x=4{\text{e}}^{-2t}$
4. Obtain the general solution of $y^{″}-y^{\prime }-2y=6$
5. Obtain the general solution of the equation $\frac{d^{2}y}{d{x}^2}+3\frac{dy}{dx}+2y=10cos2x$

   Find the particular solution satisfying $y\left(0\right)=1,\frac{dy}{dx}\left(0\right)=0$

6. Find a particular integral for the equation $\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}+y=1+x$
7. Find the general solution of
   1. $\frac{d^{2}x}{d{t}^2}-6\frac{dx}{dt}+5x=3$
   2. $\frac{d^{2}x}{d{t}^2}-2\frac{dx}{dt}+x={\text{e}}^t$

Answer