Module 2: Practice Problems
Instructions: 10 problems covering separable, linear, exact, and substitution methods.
1. Separable
Solve dy/dx = x/y, y(0) = 2.
Solution
y dy = x dx. y²/2 = x²/2 + C. y(0)=2: 2 = C. y² = x² + 4. y = sqrt(x² + 4) (positive root).
2. Separable
Solve dy/dx = y cos x.
Solution
dy/y = cos x dx. ln|y| = sin x + C. y = Aesin x.
3. Linear
Solve y' + 3y = 6, y(0) = 0.
Solution
μ = e3x. d/dx[e3xy] = 6e3x. e3xy = 2e3x+C. y = 2+Ce-3x. y(0)=0: C=-2. y = 2 - 2e-3x.
4. Linear
Solve xy' + 2y = x³, x > 0.
Solution
Standard form: y' + (2/x)y = x². μ = x². d/dx[x²y] = x4. x²y = x5/5 + C. y = x³/5 + C/x².
5. Exact
Solve (2x + y²) dx + (2xy + 1) dy = 0.
Solution
∂M/∂y = 2y = ∂N/∂x. Exact. F = x² + xy² + g(y). ∂F/∂y = 2xy + g' = 2xy + 1, so g' = 1, g = y. x² + xy² + y = C.
6. Exact
Solve (3x²y + ey) dx + (x³ + xey) dy = 0.
Solution
∂M/∂y = 3x² + ey = ∂N/∂x. Exact. F = x³y + xey + g(y). ∂F/∂y = x³ + xey + g' = x³ + xey. g' = 0, g = 0. x³y + xey = C.
7. Bernoulli
Solve y' - y = -y³.
Solution
n=3, v = y-2. Divide by y³: y-3y' - y-2 = -1. -v'/2 - v = -1. v' + 2v = 2. μ = e2x. v = 1 + Ce-2x. y² = 1/(1 + Ce-2x).
8. Bernoulli
Solve y' + y/x = y² ln x, x > 0.
Solution
n=2, v = 1/y. v' - v/x = -ln x. μ = 1/x. d/dx[v/x] = -ln(x)/x. v/x = -(ln x)²/2 + C. 1/y = x[C - (ln x)²/2].
9. Homogeneous
Solve dy/dx = (y² + xy)/x².
Solution
= (y/x)² + y/x = v² + v. v + xv' = v² + v, so xv' = v². dv/v² = dx/x. -1/v = ln|x| + C. v = -1/(ln|x|+C). y = -x/(ln|x| + C).
10. Identify and Solve
Solve (1 + ex/y) dx + ex/y(1 - x/y) dy = 0. (Hint: check if homogeneous.)
Solution
Let v = x/y (homogeneous-type). Alternatively, note M and N satisfy ∂M/∂y = -(x/y²)ex/y and ∂N/∂x = (1/y)ex/y(1-x/y+1) -- checking reveals exactness after manipulation. The potential function is F = x + yex/y. Solution: x + yex/y = C.