Question

The general solution to the differential equation $\frac{dy}{dx}=2xy$ can be represented as:

• A. $y=Ke^{x^2}$
• B. $y=Kx$
• C. $y=Kx^2$
• D. $y=Cx^2$

Answer 1

Answer: (A)

$y=Ke^{x^2}$

Answer 2

The given differential equation is a first-order separable ordinary differential equation. Separate variables to get $\frac{dy}{y} = 2x \,dx$. Integrate both sides: $\int \frac{dy}{y} = \int 2x \,dx$, which yields $\ln|y| = x^2 + C$. Exponentiate both sides to solve for $y$: $|y| = e^{x^2+C} = e^{x^2}e^C$. Let $K = \pm e^C$ (allowing for $K=0$ to include the trivial solution $y=0$), resulting in the general solution $y=Ke^{x^2}$.