Question

home / study / math / advanced math / advanced math questions and answers / find the solution of ivp for the differential equation (x+2)^2e^y when y(1) =0 y'= dy/dx=(x+2)^2.e^y;... Question: Find the solution of IVP for the differential equation (x+2)^2e^y when y(1) =0 y'= dy/dx=(x+2)^2.... Find the solution of IVP for the differential equation (x+2)^2e^y when y(1) =0 y'= dy/dx=(x+2)^2.e^y; y(1)=0

Answer 1

`y=-\ln\left(-((x+2)^3)/(3)+10\right)`

Explanation:

Using separation of variables we have
`y'=(dy)/(dx)=(x+2)^2e^y \Rightarrow e^(-y)dy=(x+2)^2dx \Rightarrow \int e^(-y)dy=\int(x+2)^2dx \Rightarrow - e^(-y)=((x+2)^3)/(3)+k`, using the initial condition
`y(1)=0`, we obtain
`-e^(-0)=((1+2)^3)/(3)+k \Rightarrow k=-e^(-0)-3^2=-1-9=-10` and clearing
`y` it follows
`-y=\ln e^(-y)=\ln \left(-((x+2)^3)/(3)+10\right) \Rightarrow y=-\ln\left(-((x+2)^3)/(3)+10\right)`.