General And Particular Solutions Of A Differential Equation

The equation of the family of circles of radius r is

(x%20-%20a)%5E2%20%2B%20(y%20-%20b)%5E2%20%3D%20r%5E2

%5Cquad

%5Cquad

%5Cquad

...(1)
Where a & b are arbitrary constants.
Since equation (1) contains two arbitrary constants, we differentiate it two times w.r.t x & the differential equation will be of second order.
Differentiating (1) w.r.t. x, we get

2%5Cleft(x-a%5Cright)%2B2%5Cleft(y-b%5Cright)%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D0

%5CRightarrow%20%5Cleft(x-a%5Cright)%2B2%5Cleft(y-b%5Cright)%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D0%5Cquad%5Cquad%5Cquad%5Cquad...%5Cleft(2%5Cright)

Differentiating (2) w.r.t. x, we get

1%2B%5Cleft(y-b%5Cright)%20%5Cfrac%7Bd%5E%7B2%7Dy%7D%7Bdx%5E%7B2%7D%7D%2B%5Cleft(%5Cfrac%7Bdy%7D%7Bdx%7D%5Cright)%5E%7B%5E2%7D%3D0%5Cquad%5Cquad%5Cquad%5Cquad...%5Cleft(3%5Cright)

%5CRightarrow%20%5Cleft(y-b%5Cright)%3D-%5Cfrac%7B1%2B%5Cleft(%5Cfrac%7Bdy%7D%7Bdx%7D%5Cright)%5E%7B%5E2%7D%7D%7B%5Cfrac%7Bd%5E%7B2%7Dy%7D%7Bdx%5E%7B2%7D%7D%7D%5Cquad%5Cquad%5Cquad%5Cquad%5Cquad...%5Cleft(4%5Cright)

On putting the value of (y - b) in equation(2), we get

x-a%3D%5Cfrac%7B%5Cleft%5B1%2B%5Cleft(%5Cfrac%7Bdy%7D%7Bdx%7D%5Cright)%5E%7B%5E2%7D%5Cright%5D%20%5Cfrac%7Bdy%7D%7Bdx%7D%7D%7B%5Cfrac%7Bd%5E%7B2%7Dy%7D%7Bdx%5E%7B2%7D%7D%7D%5Cquad%5Cquad%5Cquad%5Cquad%5Cquad...%5Cleft(5%5Cright)

Substituting the values of (x - a) &(x - b)in (1), we get

%5Cfrac%7B%5Cleft%5B1%2B%5Cleft(%5Cfrac%7Bdy%7D%7Bdx%7D%5Cright)%5E%7B%5E2%7D%5Cright%5D%5E%7B%5E%7B%5E2%7D%7D%5Cleft(%5Cfrac%7Bdy%7D%7Bdx%7D%5Cright)%5E%7B%5E2%7D%7D%7B%5Cleft(%5Cfrac%7Bd%5E%7B2%7Dy%7D%7Bdx%5E%7B2%7D%7D%5Cright)%5E%7B%5E2%7D%7D%2B%5Cfrac%7B%5Cleft%5B1%2B%5Cleft(%5Cfrac%7Bdy%7D%7Bdx%7D%5Cright)%5E%7B%5E2%7D%5Cright%5D%5E%7B%5E%7B%5E2%7D%7D%7D%7B%5Cleft(%5Cfrac%7Bd%5E%7B2%7Dy%7D%7Bdx%5E%7B2%7D%7D%5Cright)%5E%7B%5E2%7D%7D%3Dr%5E%7B2%7D

%5CRightarrow%20%5Cleft%5B1%2B%5Cleft(%5Cfrac%7Bdy%7D%7Bdx%7D%5Cright)%5E%7B%5E2%7D%5Cright%5D%5E%7B%5E%7B%5E3%7D%7D%3Dr%5E%7B2%7D%5Cleft(%5Cfrac%7Bd%5E%7B2%7Dy%7D%7Bdx%5E%7B2%7D%7D%5Cright)%5E%7B%5E2%7D

The given differential equation is

xdy-ydx-%5Csqrt%7Bx%5E%7B2%7D%2By%5E%7B2%7D%7Ddx%3D0

%5CRightarrow%5Cquad%20xdy%3D%5Cleft(y%2B%5Csqrt%7Bx%5E%7B2%7D%2By%5E%7B2%7D%7D%5Cright)dx

%5CRightarrow%20%5Cquad%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7B%5Cleft(y%2B%5Csqrt%7Bx%5E%7B2%7D%2By%5E%7B2%7D%7D%5Cright)%7D%7Bx%7D

This is the linear differential equation.
Put y = vx, so that

%5Cfrac%7Bdy%7D%7Bdx%7D%3Dv%2Bx%5Cfrac%7Bdy%7D%7Bdx%7D.

Then

v%2Bx%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7Bvx%2B%5Csqrt%7Bx%5E%7B2%7D%2Bv%5E%7B2%7Dx%5E%7B2%7D%7D%7D%7Bx%7D

%5CRightarrow%5Cquad%20v%2Bx%5Cfrac%7Bdy%7D%7Bdx%7D%3Dv%2B%5Csqrt%7B1%2Bv%5E%7B2%7D%7D

%5Cquad%5Cquad%5Cquad%5Cfrac%7Bdv%7D%7B%5Csqrt%7B1%2Bv%5E%7B2%7D%7D%7D%3D%5Cfrac%7Bdx%7D%7Bx%7D

Integrating both sides, we get

%5Cint%5Cfrac%7Bdv%7D%7B%5Csqrt%7B1%2Bv%5E%7B2%7D%7D%7D%3D%5Cint%5Cfrac%7Bdx%7D%7Bx%7D

%5CRightarrow%5Cquad%20log%5Cleft(v%2B%5Csqrt%7B1%2Bv%5E%7B2%7D%7D%5Cright)%3Dlog%20x%20%2B%20log%20C

%5CRightarrow%20%5Cquad%20v%2B%5Csqrt%7B1%2Bv%5E%7B2%7D%7D%3DCx

%5CRightarrow%20%5Cquad%20%5Cfrac%7By%7D%7Bx%7D%2B%5Csqrt%7B1%2B%5Cfrac%7By%5E%7B2%7D%7D%7Bx%5E%7B2%7D%7D%7D%3DCx%5Cquad%5Cleft%5B%5Cbecause%20y%3Dvx%5Cright%5D

y%2B%5Csqrt%7Bx%5E%7B2%7D%2By%5E%7B2%7D%7D%3DCx%5E%7B2%7D

About General And Particular Solutions Of A Differential Equation - VITEEE

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General And Particular Solutions Of A Differential Equation

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About General And Particular Solutions Of A Differential Equation - VITEEE

Frequently Asked Questions

Why focus on General And Particular Solutions Of A Differential Equation PYQs?

How to best use this analysis?

General And Particular Solutions Of A Differential Equation

High-Yield Trend

Chapter Questions 3 MCQs

Official Solution

Official Solution

Official Solution

About General And Particular Solutions Of A Differential Equation - VITEEE

Frequently Asked Questions

Why focus on General And Particular Solutions Of A Differential Equation PYQs?

How to best use this analysis?

Chapter Questions
3 MCQs