# Question about Differential Equations.....!Please derive the solution of first order linear Differential Equation...? (Or) Show that the Solution of dy/dx+P(x)y = Q(x) is y.e^Integral of P(x)dx =...

Question about Differential Equations.....!

Please derive the solution of first order linear Differential Equation...?

(Or)

Show that the Solution of dy/dx+P(x)y = Q(x) is y.e^Integral of P(x)dx = (Integral of Q(x). e^Integral of P(x)dx ) dx + c

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### 1 Answer

Multiplicate the equation by an unknown function v(x).

`v(x)*dy/dx+v(x)*P(x)y = v(x)*Q(x)`

You may think at the term v(x)*dy/dx as being obtained from `(d(vy))/dx.`

Use product rule: `d(vy)/dx = y*(dv/dx) + v*(dy/dx)`

Comparing the equations `v(x)*dy/dx+v(x)*P(x)y and y*(dv/dx) + v* (dy/dx) =gt v(x)*P(x) = dv/dx =gt dv/v(x) = P(x)*dx`

Integrate both sides:

`int dv/v(x) = int P(x)*dx`

`` `ln|v| =int P(x)*dx`

Solve for v => `v = e^(int P(x)*dx)`

What you've just get is known as integrating factor.

You will use this factor to solve the differential equation.

`d(vy)/dx =v(x)*Q(x)`

Integrate both sides

`v(x)*y = int v(x)*Q(x) dx`

Solve for`y =gt y = (int v(x)*Q(x) dx)/(v(x))`

**ANSWER: `y = (int v(x)*Q(x) dx)/(e^(int P(x)*dx))` **