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

Answer:

$y=-\ln\left(-\frac{(x+2)^3}{3}+10\right)$

Step-by-step explanation:

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