Differential Equations
dy/dx + 2xy=x
hint: this problem is in the form of dy/dx +p(x)y=Q(x)
find integrating factor (IF) then use general formula i.e y*IF= integration of IF *Q(x)
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.