Differential Equations

Use this for “A pond can hold 2000 fish. It is stocked with 200 and grows by dP/dt = 0.3P(1 − P/2000), t in years. How many fish are there after 5 years?”

Use this for “Use Euler’s method with h = 0.25 to estimate y(1) for y′ = 2x − y, y(0) = 3. How far is it from the exact value?”

Use this for “A 400 L tank of fresh water takes in brine with 25 g/L of salt at 8 L/min, and the mixture drains at the same rate. How much salt is in it after 30 minutes?”

Use this for “Solve y′ + 3y = 6x + 1 with y(0) = 2, and find y(0.5).”

Use this for “Is y′ = y cos x separable, linear, or both? Which method solves it?”

Use this for “Find the general solution of y″ − 5y′ + 6y = 0.”

Use this for “For y″ + 0.5y′ + 4y = cos(ωt), how large is the steady swing at ω = 2, and at ω = 1?”

Use this for “Find the Laplace transform of 3t²e²ᵗ, and its value at s = 4.”

Use this for “Find the inverse Laplace transform of (s + 3) ÷ ((s + 1)(s + 2)), and its value at t = 1.”

Use this for “Solve y′ + 2y = 6, y(0) = 1, by the Laplace transform.”

Use this for “Solve x′ = x + 2y, y′ = 2x + y with x(0) = 3, y(0) = 1. Where is the solution at t = 0.5?”

Use this for “Rabbits x and foxes y, in hundreds, follow x′ = x − 0.5xy, y′ = −0.75y + 0.25xy. Where do both stay constant, and how long is one small cycle?”

Use this for “What kind of equilibrium does x′ = Ax have at the origin when A = [[−1, −2], [2, −1]]?”

Use this for “Find the first terms of the series solution of y″ = xy with y(0) = 1, y′(0) = 0, and estimate y(1).”

Use this for “A 1 m rod with its ends at 0 °C starts at u = 100 sin(πx). With k = 0.01 m²/s, find u at x = 0.5 m after 10 s.”