a solve ODE by seperation of variables b find the particular

a) solve ODE by seperation of variables

b) find the particular solution if y(0)=0

(1/x^2 e^x) dy = (1 + y^2) dx solve ODE by seperation of variables find the particular solution if y(0)=0

Solution

given (1/(x²e^x)) dy =(1+y²) dx

seperate variables

dy/(1+y²) =x²e^x dx

integrate on both sides

dy/(1+y²) =x²e^x dx

tan^-1(y)=x²e^x dx

integration by parts:u=x²,du =2xdx , dv =e^xdx,v=e^x

udv =uv-vdu

tan^-1(y)=x²e^x - e^x2xdx

tan^-1(y)=x²e^x - 2xe^xdx

integration by parts:u=x,du =dx , dv =e^xdx,v=e^x

udv =uv-vdu

tan^-1(y)=x²e^x - 2[xe^x- e^xdx]

tan^-1(y)=x²e^x - 2[xe^x- e^x] +c

tan^-1(y)=x²e^x - 2xe^x+2e^x+c

y=tan(x²e^x - 2xe^x+2e^x+c)

b)y(0)=0

0=0²e⁰ - 2*0e⁰+2e⁰+c

0=0-0+2+c

c=-2

y=tan(x²e^x - 2xe^x+2e^x-2) is particular solution