10 questions covering all Module 1 topics. Work each problem, then reveal the answer to check.
1
What is the order and linearity of: y''' + 4y' = ex?
Answer: Third order, linear. y and its derivatives appear to the first power with coefficients depending only on x (constants here).
2
Is the equation y y'' + (y')² = 0 linear or nonlinear? Why?
Answer: Nonlinear. The term yy'' is a product of y and its second derivative, and (y')² squares the first derivative.
3
Verify that y = Ce5x is a general solution to y' = 5y.
Answer: y' = 5Ce5x. Check: y' = 5y gives 5Ce5x = 5(Ce5x). True for all C. Verified.
4
Solve the IVP: dy/dx = cos x, y(π/2) = 1.
Answer: y = sin x + C. Apply y(π/2) = 1: 1 = sin(π/2) + C = 1 + C, so C = 0. Answer: y = sin x.
5
For dy/dx = y(2 - y), find all equilibrium solutions and classify each as stable or unstable.
Answer: Equilibria: y = 0 and y = 2. For 0 < y < 2, f > 0 (increasing); for y > 2, f < 0 (decreasing); for y < 0, f < 0 (decreasing). So y = 2 is stable, y = 0 is unstable.
6
For the direction field of dy/dx = x² - y, what is the slope at (1, 2)?
Answer: Slope = 1² - 2 = -1.
7
Does the IVP dy/dx = √(|y|), y(0) = 0 satisfy the uniqueness condition?
Answer: No. df/dy is undefined at y = 0 (the initial value). Uniqueness is not guaranteed -- and indeed this IVP has multiple solutions.
8
Use one step of Euler's method (h = 0.5) for dy/dx = -3y, y(0) = 2. Find y(0.5).
Answer: f(0, 2) = -3(2) = -6. y1 = 2 + 0.5(-6) = 2 - 3 = -1. (Note: exact y(0.5) = 2e-1.5 ≈ 0.446. The large step causes the Euler estimate to overshoot into negative territory.)
9
How many arbitrary constants appear in the general solution of a second-order ODE?
Answer:Two. An n-th order ODE has n arbitrary constants in its general solution.
10
The solution to dy/dx = 1 + y², y(0) = 0 is y = tan x. On what interval does this solution exist?
Answer: y = tan x has vertical asymptotes at x = ±π/2. The solution exists on (-π/2, π/2). Despite f being smooth, the solution blows up at finite x.