# Solve differential equationdy/dx= 15x^2 e^-y

First order differential equations

Solve differential equation $$dy/dx= 15x^2 e^{-y}$$

2020-11-06

$$dy/dx= 15x^2 e^{-y}$$
$$e^ydy= 15x^2dx$$ (1)
On integrating equation (1) we get
$$\int e^{y}dy= 15 \int x^{2}dx$$
$$\Rightarrow e^{y}=5x^{3}+C$$ (2)
Take logarithmic on the both side if equation (2) we get
$$\ln(e^y)= \ln(abs(5x^3+c))$$
$$\Rightarrow y \ln(e)=\ln(abs(5x^3+c))$$
$$\Rightarrow y= \ln(abs(5x^3+c))$$
$$y(x)=\ln(abs(5x^3+c))$$