Differential Equations?

Question

Find a 1-paramter family of solutions of each of the following equations. Assume in each case that the coefficient of dy DOES NOT EQUAL ZERO.

There are 3 problems:

1.)	xy’ – y – xsin(y/x) = 0

2.)	(2x2 y + y3)dx + (xy2 – 2x3)dy = 0

3.)	y2 dx + [x√(y2 - x2) – xy]dy = 0

The answers are:

1.)	y = 2x[arctan(cx)]

2.)	c = x2 / y2  + log(xy)

3.)	y2 – cx = y√(y2 - x2)


Any help is appreciated on how to solve through these. I have gotten several diff answers for each one and each time have been incorrect.

Learner · Accepted Answer

All the three problems given here are coming under homogeneous type; This require a standard substitution in the form y = vx; ==> (dy/dx) = v + x(dv/dx)

Question-1:

1) substituting as given above, we get
vx + x^2(dv/dx) - vx - x sin(v) = 0

2) ==> (dv/dx) = sin(v)/x
3)==> dv/sin(v) = dx/x

4) Integrating both sides, log{tan(v/2)} = log(x) + log(c)

5) ==> log{tan(v/2)} = log(cx)
6) ==> tan(v/2) = cx
7) ==> v/2 = Arc tan(cx), substituting v = y/x,

The answer is; y = 2x[arc tan(cx)}

Question-2:

1) dy/dx = (2x^2 y + y^3)/(2x^3 - xy^2)

You may try with the same substitution as given above and the get the end result;

Question-3:
 1) dy/dx = (y^2) / [xy - x√(y2 - x2]; This also you may try as above and arrive at the end result,

This is now already half past midnight; hence it may not be possible to provide more than the above; however, if possible I shall try to present the detailed work out for the other two tomorrow; in the mean time you may make an attempt of your own.

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Differential Equations?

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