Finding the general solution of a second order linear inhomogeneous ODE

5 Finding the general solution of a second order linear inhomogeneous ODE

The general solution of a second order linear inhomogeneous equation is the sum of its particular integral and the complementary function. In subsection 2 (page 32) you learned how to find a complementary function, and in subsection 4 (page 42) you learnt how to find a particular integral. We now put these together to find the general solution.

Example 15

Find the general solution of $\frac{d^{2} y}{d x^{2}} + 3 \frac{d y}{d x} - 10 y = 3 x^{2}$

Solution

The complementary function was found in Example 8 to be $y_{cf} = A e^{2 x} + B e^{- 5 x}$ .

The particular integral is found by trying a solution of the form $y = a x^{2} + b x + c$ , so that

$\frac{d y}{d x} = 2 a x + b, \frac{d^{2} y}{d x^{2}} = 2 a$

Substituting into the differential equation gives

$2 a + 3 (2 a x + b) - 10 (a x^{2} + b x + c) = 3 x^{2}$

Comparing constants: $2 a + 3 b - 10 c = 0$

Comparing $x$ terms: $6 a - 10 b = 0$

Comparing $x^{2}$ terms: $- 10 a = 3$

So $a = - \frac{3}{10}$ , $b = - \frac{9}{50}$ , $c = - \frac{57}{500}$ , $y_{p} (x) = - \frac{3}{10} x^{2} - \frac{9}{50} x - \frac{57}{500}$ .

Thus the general solution is $y = y_{p} (x) + y_{cf} (x) = - \frac{3}{10} x^{2} - \frac{9}{50} x - \frac{57}{500} + A e^{2 x} + B e^{- 5 x}$

Key Point 9

The general solution of a second order constant coefficient ordinary differential equation

$a \frac{d^{2} y}{d x^{2}} + b \frac{d y}{d x} + c y = f (x) is y = y_{p} + y_{cf}$

being the sum of the particular integral and the complementary function.

$y_{p}$ contains no arbitrary constants; $y_{cf}$ contains two arbitrary constants.