Question

What is the solution to the differential equation $\frac{dy}{dx}=5y^2$ with the initial condition $y\left(0\right)=3$?

Answer 1

Solution of differential equation:

Given,

$\frac{dy}{dx}=5y^2$ and $y\left(0\right)=3$

on solving the differential equation,

$$\begin{gathered} \frac{dy}{dx}=5y^2 \\ \Rightarrow \frac{dy}{y^2}=5dx \\ \Rightarrow \int \frac{1}{y^2}dy=\int 5dx \\ \Rightarrow \frac{-1}{y}=5x+c \end{gathered}$$

substituting $y\left(0\right)=3$ ,in the above equation we get

$$\begin{gathered} \Rightarrow \frac{-1}{3}=5\left(0\right)+c \\ \Rightarrow c=\frac{-1}{3} \end{gathered}$$

Now the required equation will be

$\frac{1}{y}+5x-\frac{1}{3}=0$

Hence,$\frac{1}{y}+5x-\frac{1}{3}=0$ is the required solution.