General And Particular Solutions Of A Differential Equation

Given equation is

%5Csin%20x%5Ccos%203x%3D%5Csin%203x%5C%2C%5Ccos%20%5C%2C5x

%5CRightarrow

2%5Csin%20x%5C%2C%5Ccos%20%5C%2C3x-2%5Csin%203x%5C%2Ccos5x%3D0

%5CRightarrow

%5Csin%20(3x%2Bx)-%5Csin%20(3x-x)-%5Csin%20(3x%2B5x)

%2B%5Csin%20(5x-3x)%3D0

%5CRightarrow

%5Csin%204x-%5Csin%202x-%5Csin%208x%2B%5Csin%202x%3D0

%5CRightarrow

%5Csin%204x-%5Csin%208x%3D0

%5CRightarrow

2%5Ccos%20%5Cleft(%20%5Cfrac%7B4x%2B8x%7D%7B2%7D%20%5Cright)%5Csin%20%5Cleft(%20%5Cfrac%7B8x-4x%7D%7B2%7D%20%5Cright)%3D0

%5CRightarrow

2%5Ccos%20%5C%2C6x%5C%2C%5Csin%20%5C%2C2x%3D0

%5CRightarrow

%5Ccos%20%5C%2C%5C%2C6x%3D0

%5Csin%20%5C%2C2x%3D0

%5CRightarrow

6x%3D(2n%2B1)%5Cfrac%7B%5Cpi%20%7D%7B2%7D

x%3D%5Cfrac%7Bn%5Cpi%20%7D%7B2%7D

%5CRightarrow

x%3D(2n%2B1)%5Cfrac%7B%5Cpi%20%7D%7B12%7D

x%3D%5Cfrac%7Bn%5Cpi%20%7D%7B2%7D

%5CRightarrow

x%3D0%2C%5C%2C%5Cfrac%7B%5Cpi%20%7D%7B2%7D%2C%5Cfrac%7B%5Cpi%20%7D%7B12%7D%2C%5Cfrac%7B3%5Cpi%20%7D%7B12%7D%2C%5Cfrac%7B5%5Cpi%20%7D%7B12%7D%5Cin%20%5Cleft%5B%200%2C%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%5Cright%5D

%5Ctherefore

Number of solutions is 5.

We have,

%5Cfrac%7Bdy%7D%7Bdx%7D%3D(y%2B4x-4)%5E%7B2%7D

put

y%2B4x%3Du

so that

%5Cfrac%7Bdt%7D%7Bdx%7D%2B4%3D%5Cfrac%7Bdu%7D%7Bdx%7D

So, the given differential equation becomes

%5Cfrac%7Bdu%7D%7Bdx%7D-4

%3D(u-4)%5E%7B2%7D

%5CRightarrow%20%5Cfrac%7Bdu%7D%7Bdx%7D

%3D(u%2B4)%5E%7B2%7D%2B4

%5CRightarrow%20%5Cfrac%7Bdu%7D%7B%5Cleft(u-4%5Cright)%5E%7B2%7D%2B%5Cleft(2%5Cright)%5E%7B2%7D%7D%3Ddx

On integrating, we get

%5Cint%20%5Cfrac%7Bdu%7D%7B%5Cleft(u-4%5Cright)%5E%7B2%7D%2B%5Cleft(2%5Cright)%5E%7B2%7D%7D%3D%5Cint%20dx%20%2Bc

%5CRightarrow%20%5Cfrac%7B1%7D%7B2%7D%20tan%5E%7B-1%7D%5Cleft(%5Cfrac%7Bu-4%7D%7B2%7D%5Cright)%3Dx%2Bc

%5CRightarrow%20%5Cfrac%7B1%7D%7B2%7Dtan%5E%7B-1%7D%20%5Cleft(%5Cfrac%7By%2B4x-4%7D%7B2%7D%5Cright)

%3Dx%2Bc%20%5Cleft%5B%5Cbecause%20u%3Dy%2B4x%5Cright%5D

When

x%20%3D%200

y%3D4

, we have

%5Cfrac%7B1%7D%7B2%7D%20tan%5E%7B-1%7D%5Cleft(0%5Cright)%3D0%2Bc

%5CRightarrow%20c%3D0

%5Ctherefore

Solution is,

%5Cfrac%7B1%7D%7B2%7D%20tan%5E%7B-1%7D%20%5Cleft(%5Cfrac%7By%2B4x-4%7D%7B2%7D%5Cright)%3Dx

%5CRightarrow%20y%2B4x-4%3D2

tan

%5Cleft(2x%5Cright)

, which is required solution

Chapter Questions
3 MCQs