Answer to Question #302573 in Differential Equations for haru

Question

Solve the following initial value problems using method of undetermined coefficients: (i) y−9y = ex+x−1, y(0) = −1,y(0) = 1,

Answer 1

Correponding homogeneous differential equation

$$y''-9y=0$$

Characteristic (auxiliary) equation

$$r^2-9=0$$

$$r_1=3, r_2=-3$$

The general solution of the homogeneous differential equation is

$$y_h=c_1e^{3x}+c_2e^{-3x}$$

Find the particular solution of the non homogeneous differential equation

$$y_p=Ae^x+Bx+C$$

$$y_p'=Ae^x+B$$

$$y_p''=Ae^x$$

Substitute

$$Ae^x-9Ae^x-9Bx-9C=e^x+x-1$$

$$A=-1/8$$

$$B=-1/9$$

$$C=1/9$$

The general solution of the non homogeneous differential equation is

$$y=c_1e^{3x}+c_2e^{-3x}-\dfrac{1}{8}e^x-\dfrac{1}{9}x+\dfrac{1}{9}$$

$y'(0) = −1,y(0) = 1$

$$3c_1-3c_2-\dfrac{1}{8}-\dfrac{1}{9}=-1$$

$$c_1+c_2-\dfrac{1}{8}+\dfrac{1}{9}=1$$

$$c_2=\dfrac{73}{72}-c_1$$

$$6c_1-\dfrac{219}{72}=-\dfrac{55}{72}$$

$$c_1=\dfrac{82}{216}$$

$$c_2=\dfrac{137}{216}$$

The solution of the given IVP is

$$y=\dfrac{82}{216}e^{3x}+\dfrac{137}{216}e^{-3x}-\dfrac{1}{8}e^x-\dfrac{1}{9}x+\dfrac{1}{9}$$