Notes On Linear Differential Equations - CBSE Class 12 Maths
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

#### 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

Next