Solving Math Problems (part 4)

Here are the next problems in Erwin Kreyszig's Advanced Engineering Mathematics, 6th edition.

Section 1.2, Problem 12

Solve the differential equation

Separating variables gives:

Integrating both sides gives:

The left-hand side is solved from the table of integrals in the front cover of the textbook.

arc tan y = 𝜋 x + C

y = tan ( 𝜋 x + C)

Section 1.2, Problem 13

Solve the differential equation

Separating variables gives:

Integrating both sides gives:

The left hand side is solved using an identity from the front cover of the text book.

arc sin y = x + C

y = sin (x + C)

Section 1.2, Problem 14

Solve the differential equation

y' + y² = 1

Separating the variables gives:

Integrating both sides gives:

The left hand side looks like it should be a trig identity, but it is not, it is solved via trigonometric substitution, but using Jan J. Tuma's Engineering Mathematics Handbook, 3rd Edition, Section 19, table 10 tells us that the solution to the left hand side is:

This can be manipulated over several steps to result in: