Question

The indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution y2(x) of the homogeneous equation and a particular solution yp(x) of the given nonhomogeneous equation. y'' + y' = 1; y1 = 1

Answer 1

Answer:

Required solution is $y(x)=1+A\cos x+B\sin x$ where A and B are constants.

Step-by-step explanation:

Given nonhomogenous differential equation is,

$y''(x)+y'(x)=1\hfill (1)$   with $y_1(x)=1$

To find another solution, consider $m=\frac{\partial}{\partial x}$ be such that,

$m^2+1=0\implies m=\pm i$

Hence,

$C.F=A\cos x+B\sin x$   where A and B are constants.

Let  $D=\frac{\partial}{\partial x}$

$P.I=\frac{1}{1+D^2}(1)$

$=(1+D^2)^{-1}(1)$

$=(1-D^2+.....)(1)=1$

Hence,

$y(x)=1+A\cos x+B\sin x$ where A and B are constants.

which is required solution.