Question about Differential Equations.....!
Please derive the solution of first order linear Differential Equation...?
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
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.
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))`