Solving Math Problems (Part 13)

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

Section 1.4, Problem 7

Find the general solution of

xy' = y + x² sec (y/x)

Dividing by x gives:

y' = (y/x) + x sec (y/x)

Substituting u = y/x and y' = u + u'x we get:

u + u'x = u + x sec u

Re-arranging we get:

∫cos u du = ∫ dx

sin u = x + C

u = arc sin (x + C)

y = x arc sin (x + C)

Section 1.4, Problem 8

Find the general solution of

Substituting u = y/x and y' = u + u'x we get:

u(u + u'x) = u² + x/(x²+1)

Simplifying we get:

uu' = 1/(x² + 1)

Integrating both sides, we have:

u²/2 = arc tan x + C

Substituting y = ux, we have: