What is a solution to the differential equation #dy/dx=x^2y#?

First, separate the variables:

Integrate both sides:

The solution to the differential equation dy/dx = x^2y is y = Ce^(x^3/3), where C is a constant.

A solution to the differential equation dy/dx = x^2y is a function y(x) that satisfies the given equation. One common method to solve this type of first-order ordinary differential equation is by using separation of variables.

First, we rewrite the equation in the form dy/y = x^2 dx. Then, we integrate both sides of the equation with respect to their respective variables:

∫(1/y) dy = ∫x^2 dx

Integrating both sides gives:

ln|y| = (1/3)x^3 + C

Where C is the constant of integration. To solve for y, we exponentiate both sides:

|y| = e^(x^3/3 + C)

Since e^C is just another constant, we can rewrite the equation as:

|y| = Ce^(x^3/3)

Where C is a constant. However, since we're solving for y, we typically ignore the absolute value sign and write:

y = Ce^(x^3/3)

Where C is an arbitrary constant. This equation represents the general solution to the given differential equation.