Module 2 Quiz
Test Your Knowledge
10 questions on separable, linear, exact, and substitution methods.
1
Solve dy/dx = 3x²e-y.
Separable: eydy = 3x²dx. ey = x³ + C. y = ln(x³ + C).
2
Solve y' - 4y = 8, y(0) = 0.
μ = e-4x. y = -2 + Ce4x. y(0)=0: C=2. y = -2 + 2e4x.
3
Is (y cos x + sin y) dx + (sin x + x cos y) dy = 0 exact?
∂M/∂y = cos x + cos y. ∂N/∂x = cos x + cos y. Yes, exact.
4
Solve the exact equation from Q3.
F = ∫(y cos x + sin y)dx = y sin x + x sin y + g(y). ∂F/∂y = sin x + x cos y + g' = sin x + x cos y. g' = 0. y sin x + x sin y = C.
5
Solve dy/dx = (x + y)/x using a homogeneous substitution.
v = y/x. v + xv' = 1 + v. xv' = 1. v = ln|x| + C. y = x ln|x| + Cx.
6
What substitution transforms y' + y = xy² into a linear ODE?
Bernoulli with n=2. v = y-1 = 1/y.
7
Solve dy/dx = (y - x)/(y + x) using v = y/x.
RHS = (v-1)/(v+1). v + xv' = (v-1)/(v+1). xv' = (v-1)/(v+1) - v = (v-1-v²-v)/(v+1) = -(1+v²)/(v+1). Separate: (v+1)/(1+v²)dv = -dx/x. ∫ = (1/2)ln(1+v²) + arctan(v) = -ln|x| + C. Substitute back v = y/x.
8
Solve y' + (1/x)y = 1/x², x > 0.
μ = x. d/dx[xy] = 1/x. xy = ln x + C. y = (ln x + C)/x.
9
Solve (2xy + 1) dx + (x² + 3y²) dy = 0.
∂M/∂y = 2x = ∂N/∂x. Exact. F = x²y + x + g(y). ∂F/∂y = x² + g' = x² + 3y². g' = 3y², g = y³. x²y + x + y³ = C.
10
Classify and state the method: (a) dy/dx = y/x, (b) y' + 2xy = x, (c) y' = y + y².
(a) Separable (also homogeneous). (b) Linear -- integrating factor. (c) Separable; also Bernoulli with n=2.