Solving Math Problems (Part 11)

More problems from Advanced Engineering Mathematics, 6th Edition by Kreyszig.

Section 1.4, problem 3

Find the general solution to:

xy' = (y - x)³ + y

Expanding the bracket and dividing everything by x³ we get:

Using a change of variable where u = y/x and y' = u + u'x we get:

Simplifying we get:

Separating the variables by cross multiplication and then integrating both sides we have:

Some re-arrangement is necessary before we can convert back to y from u.

Converting back to y from u we have:

Section 1.4, Problem 4

Find the general solution of:

xy' = y² + y

Dividing by x² we get:

Change of variable using u = y/x and y' = u + u'x gives:

This simplifies to:

u' = u²

Integrating, we get:

 Re-arranging:

 Changing back to y from u gives