Solve the differential equation dy/dx=x2y+y

1.	Solve the differential equation dy/dx=x2y+y
Answer» We're given to solve the DIFFERENTIAL equation, $\longrightarrow\dfrac{dy}{dx}=x^2y+y$ or, $\longrightarrow\dfrac{dy}{dx}=\left(x^2+1\right)y$ $\longrightarrow\dfrac{1}{y}\,dy=\left(x^2+1\right)\ dx$ Integrating, $\displaystyle\longrightarrow\int\dfrac{1}{y}\,dy=\int\left(x^2+1\right)\ dx$ $\displaystyle\longrightarrow\log\|y\|-\log\|<klux>C</klux>\|=\dfrac{x^3}{3}+x$ $\displaystyle\longrightarrow\log\left\|\dfrac{y}{C}\right\|=\dfrac{x^3}{3}+x$ TAKING antilog, $\displaystyle\longrightarrow\dfrac{y}{C}=e^{\frac{x^3}{3}+x}$ $\displaystyle\longrightarrow\underline{\underline{y=Ce^{\frac{x^3}{3}+x}}}$ This is the SOLUTION to the differential equation, where C is an arbitrary CONSTANT.

Answer»

We're given to solve the DIFFERENTIAL equation,

$\longrightarrow\dfrac{dy}{dx}=x^2y+y$