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Linear Differential Equations

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Linear Differential Equations - Lesson Summary

A linear differential equation is of the form

dy/dx + Py = Q ..... (1) Where P and Q are constants or functions of x.

First multiply both its sides by a function h(x).

h(x) dy/dx + Pyh(x) = Qh(x) ..... (2)

The function h (x), such that the right hand side is a derivative of the product of y and h(x).

h(x) dy/dx + Pyh(x) = d/dx[yh(x)]

⇒ h(x) dy/dx + Pyh(x) = h(x) dy/dx + yh'(x)

⇒ Pyh(x) = yh'(x)

⇒ P h(x) = h'(x)

⇒ P = h'(x)/h(x)         (integrate on both the sides)

⇒ ∫ P dx = ∫ h'(x)/h(x) dx

⇒ ∫ P dx = log(h(x))

⇒ h(x) = e∫Pdx

Substitute the value of h(x) in equation 2.

⇒ e∫Pdx dy/dx + Pe∫Pdx y = Q e∫Pdx
⇒ d/dx (ye∫Pdx) = Qe∫Pdx         (integrate both the sides of the equation)

⇒ d(ye∫Pdx) = Qe∫Pdxdx

⇒ ∫ d(ye∫Pdx) = ∫ Qe∫Pdxdx

⇒ ye∫Pdx = ∫ Qe∫Pdxdx

This is the required solution of the differential equation.

Integrating factor = e∫Pdx

The general solution of equations of type 1 can be obtained by the formula

y x (IF) = q x (I.F)dx

If x is replaced by y, and y is replaced by x in equation 1, then the linear differential equation

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

x(I.F) = ∫ Q(y) x (IF)dy , where IF = e∫Pdx

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