Solving Math Problems (Part 6)

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

Section 1.2, Problem 18

Solve

y' + csc y = 0

Separating the variable we get:

y' sin y = -1

Integrating both sides we get:

∫ sin y dy = -∫ dx

- cos y = -x + C

Multiplying by -1 and taking the arc cos of both sides gives:

y = arc cos (x + C)

Section 1.2, Problem 19

Solve the initial value problem:

xy' + y = 0 given y(1) = 1

Separating the variables gives:

Integrating both sides gives:

ln | y | = - ln | x | + C

Applying the initial condition of y(1) = 1, solves C = 1 and the solution is y = 1 / x.

Section 1.2, Problem 20

Solve the initial value problem:

y' = - x / y given y(2) = ⎷5

Separating the variables we get:

y y'  = - x

Integrating both sides we have:

∫ y dy = - ∫ x dx

½ y² = - ½ x² + C

Using the initial condition, solving for C we get C = 4.5 so that the solution is: