   Chapter 6.3, Problem 77E

Chapter
Section
Textbook Problem

Solving a Homogeneous Differential Equation In Exercises 77-82, solve the homogeneous differential equation in terms of x and y. A homogeneous differential equation is an equation of the form M ( x , y ) d x + N ( x , y ) d y = 0 where M and N are homogeneous functions of the same degree. To solve an equation of this form by the method of separation of variables, use the substitutions y = v x   and   d y = x   d v + v   d x . ( x + y ) d x − 2 x   d y = 0

To determine

To calculate: The expression for the general solution of the homogeneous differential equation given as, (x+y)dx2xdy=0.

Explanation

Given:

The homogeneous differential equation, (x+y)dx2xdy=0.

Formula used:

1xdx=lnx+C

Calculation:

Consider the homogeneous differential equation given as;

(x+y)dx2xdy=0

Divide both the left hand as well as right hand side by x and get,

(1+xy)dx2dy=0

Now, put y=vx and dy=xdv+vdx, then simplify to obtain,

(1+v)dx2(xdv+vdx)=0(1+v2v)dx2xdv=0(1v)dx=2xdvdxx=2dv1v …… (1)

Equation (1) is of variable separable form

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