Solving Math Problems (part 10)

More problems from Advanced Engineering Mathematics, 6th Edition by Kreyszig.  Section 1.2 has 2 more problems; #29 is asking for a proof and #30 asks to use this proof to re-work an example given in the book.  Following that, section 1.3 is about modeling and the exercises at the end are essentially physics word problems wherein the answer involves creating a differential equation and then solving an initial value problem.  I am skipping all that to go to section 1.4 which involves a change of variable to convert a given differential equation into a separable equation.

Section 1.4, problem 1

Find the general solution of:

xy' = x + y

We can re-arrange this to be:

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

u + u'x = 1 + u

With some re-arrangement we get:

Integrating:

gives:

u = ln |x| + C

Changing back to y using y = ux, we get:

y = x (ln |x| +C)

Section 1.4, problem 2

Find the general solution of:

xy' - 2y = 3x

Re-arrangement gives:

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

u + u'x - 2u = 3

Re-arranging to separate the variables we have:

Integrating both sides we get:

Solving:

ln |u+3| = ln |x| + C

Raising e to the power of both sides gives:

u + 3 = Cx

u = Cx - 3

Changing back to y using y = ux, we get:

y = Cx² - 3x