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)] â‡’ h(x) dy/dx + Pyh(x) = h(x) dy/dx + yh'(x) â‡’ Pyh(x) = yh'(x) â‡’ P h(x) = h'(x) â‡’ P = h'(x)/h(x)         (integrate on both the sides) â‡’ âˆ« P dx = âˆ« h'(x)/h(x) dx â‡’ âˆ« P dx = log(h(x)) â‡’ h(x) = eâˆ«Pdx Substitute the value of h(x) in equation 2. â‡’ eâˆ«Pdx dy/dx + Peâˆ«Pdx y = Q eâˆ«Pdx â‡’ d/dx (yeâˆ«Pdx) = Qeâˆ«Pdx         (integrate both the sides of the equation) â‡’ d(yeâˆ«Pdx) = Qeâˆ«Pdxdx â‡’ âˆ« d(yeâˆ«Pdx) = âˆ« Qeâˆ«Pdxdx â‡’ yeâˆ«Pdx = âˆ« Qeâˆ«Pdxdx This is the required solution of the differential equation. Integrating factor = eâˆ«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) = âˆ« Q(y) x (IF)dy , where IF = eâˆ«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)] â‡’ h(x) dy/dx + Pyh(x) = h(x) dy/dx + yh'(x) â‡’ Pyh(x) = yh'(x) â‡’ P h(x) = h'(x) â‡’ P = h'(x)/h(x)         (integrate on both the sides) â‡’ âˆ« P dx = âˆ« h'(x)/h(x) dx â‡’ âˆ« P dx = log(h(x)) â‡’ h(x) = eâˆ«Pdx Substitute the value of h(x) in equation 2. â‡’ eâˆ«Pdx dy/dx + Peâˆ«Pdx y = Q eâˆ«Pdx â‡’ d/dx (yeâˆ«Pdx) = Qeâˆ«Pdx         (integrate both the sides of the equation) â‡’ d(yeâˆ«Pdx) = Qeâˆ«Pdxdx â‡’ âˆ« d(yeâˆ«Pdx) = âˆ« Qeâˆ«Pdxdx â‡’ yeâˆ«Pdx = âˆ« Qeâˆ«Pdxdx This is the required solution of the differential equation. Integrating factor = eâˆ«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) = âˆ« Q(y) x (IF)dy , where IF = eâˆ«Pdx

Next
âž¤