Solving Math problems part 2

Here are more problems solved from section 1.2 in Advanced Engineering Mathematics, 6th edition by Erwin Kreyszig.  Up to taking this Differential Equations math course, I had found everything else in math to be intuitive.  This course started off that way, but later I had to work to understand the concepts.

Section 1.2, Problem 6

Solve

xy' = 5y

Rearranging we have:

Integrating we get:

ln |y| = ln |5x| + C

y = 5Cx

Section 1.2, Problem 7

Solve

y’ = 3x²y

Re-arranging we have:

y'/y = 3x²

Integrating we get:

ln |y| = x³ + C

Section 1.2, Problem 8

Solve

xy' = ny

We note that this is a general case of problem 6 where for 6, n = 5.  As such, we conclude that the solution is

y=nCx

Section 1.2, Problem 9

Solve

y' + ay + b = 0 (a ≠ 0)

By moving the ay+b to the other size and then dividing we get:

Integrating we get:

The left-hand side can be solve by u substitution where u = ay +b and du = ady, but I will use a table of integrals, specifically table (4) of section 19 from Engineering Mathematics Handbook, 3rd Edition, by Jan J. Tuma.  Solving the integrals we get:

Multiplying both sides by b and then taking e to the power of both sides, we get:

Finally, some rearrangement gives us the answer: