https://file.aiquickdraw.com/mathbot/file/1728550091253cesqdj1h.jpg

The equation shown in the image is:

\[
\frac{dy}{dx} - \frac{y}{x} = x^2 \sin(2x)
\]

This is a **first-order linear differential equation** of the form:

\[
\frac{dy}{dx} + P(x) y = Q(x)
\]

where \( P(x) = -\frac{1}{x} \) and \( Q(x) = x^2 \sin(2x) \).

### Solution Steps:

1. **Identify the integrating factor**:

The integrating factor (I.F.) for a first-order linear equation is given by:

\[
I.F. = e^{\int P(x) \, dx} = e^{\int -\frac{1}{x} \, dx} = e^{-\ln |x|} = \frac{1}{x}
\

Solve the first-order linear differential equation: dy/dx - y/x = x^2 sin(2x)

Detailed solution for solving the first-order linear differential equation dy/dx - y/x = x^2 sin(2x). Learn how to find the integrating factor and apply it to solve the equation step by step. Suitable for undergraduate mathematics students.

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