The document provides a solution to a linear first-order differential equation of the form x dy/dx - y = x^2 sin(x). It derives the integrating factor and integrates to find the solution y = -x cos(x) + Cx. The process involves manipulating the equation and integrating both sides.

1. Solve the following linear first order differential equations
x dy/dx - y = x^2 sinx
Solution
dy/dx - y/x = x sinx
We can find the integrating factor:
P(x) = -1/x
I = eint(-1/x)
I = eLnx^-1
I = 1/x
(1/x * y)' = 1/x * x sinx
Integrate both sides
y/x = -cosx +c
y = -x cos x + cx