Understanding the Problem

We’re given a first-order linear differential equation of the form:

dy/dx + p(x)y = Q(x)

where:

• p(x) = 2x
• Q(x) = x

Finding the Integrating Factor (IF)

The integrating factor is given by:

IF = e^(∫p(x)dx)

Substituting p(x) = 2x:

IF = e^(∫2xdx) = e^(x^2)

Applying the General Formula

The general solution is given by:

y * IF = ∫(IF * Q(x))dx

Substituting the values:

y * e^(x^2) = ∫(e^(x^2) * x)dx

To solve the integral on the right side, we can use substitution:

• Let u = x^2
• du = 2x dx

So the integral becomes:

(1/2) * ∫e^u du = (1/2) * e^u + C = (1/2) * e^(x^2) + C

Therefore,

y * e^(x^2) = (1/2) * e^(x^2) + C

Dividing both sides by e^(x^2):

y = (1/2) + Ce^(-x^2)

So, the solution to the differential equation is:

y = (1/2) + Ce^(-x^2)

where C is the constant of integration.