Solving Math Problems (Part 12)

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

Section 1.4, Problem 5

Find the general solution of:

x²y' = x² - xy + y²

Dividing by x² we get

Changing the variable using u = y/x and y' = u + u'x we get

u + u'x = 1 - u + u²

Rearranging we get:

Integrating both sides:



Substituting back y instead of u, we have:

Section 1.4, Problem 6

Find the general solution of:

xy' - y = 3x⁴cos²( y / x )

Rearranging we get:

y' - (y/x) = 3x³ cos²( y / x )

Changing the variable using u = y/x and y' = u + u'x we get

u + u'x - u = 3x³ cos² u

Separating the variables we have:

Integrating both sides we have:

tan u = x³ + C

u = arc tan (x³ + C)

Changing back to y gives:

y = x arc tan (x³ + C)