   Chapter 6.3, Problem 78E

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 3 + y 3 ) d x − x y 2   d y = 0

To determine

To calculate: General solution of the homogeneous differential equation (x3+y3)dxxy2dy=0.

Explanation

Given: (x3+y3)dxxy2dy=0

Calculation:

Considering the given differential equation

(x3+y3)dxxy2dy=0

Substituting y=vx gives,

dy=xdv+vdx

Now,

(x3+v3x3)dxx3v2(xdv+vdx)=0x3(1+v3)dxx4v2dvx3v3dx=0x3[dx<

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