Question

The solution for the differential equation $\frac{dy}{dx}=x^{2}y$ with the condition that y = 1 at x = 0 is

• A. y = $e^{\frac{1}{2x}}$
• B. $ln\left(y\right)=\frac{x^3}{3}+4$
• C. $ln\left(y\right)=\frac{x^2}{2}$
• D. $y=e^{\frac{x^3}{3}}$

Answer 1

The correct option is D $y=e^{\frac{x^3}{3}}$

Given, $\frac{dy}{dx}=x^{2}y;$ {variables are separable}

$\Rightarrow \frac{dy}{y}=x^{2}dx$

Integrating both sides, we get

$\int \frac{dy}{y}=\int x^{2}dx+C$

$lny=\frac{x^3}{3}+C$

Given, y(0) = 1

$ln1=0+C$

$\Rightarrow C=0$

$\Rightarrow lny=\frac{x^3}{3}$

y = $e^{\left(x^3/3\right)}$