MATH 365 Ordinary Differential Equations Exercises

The most up-to-date version of this collection of homework exercises can always be found at 1. Find the general solution to the following ordinary differential equation. dy dx = (y − 1)(x − 2)(y + 3) (x − 1)(y − 2)(x + 3) 2. Find the general solution to the following ordinary differential equation. x 3 e 2x 2 +3y 2 dx − y 3 e −x 2 −2y 2 dy = 0 3. Find the general solution to the following ordinary differential equation. dU ds = U + 1 √ s + √ sU 4. Find the solution to the following initial value problem. dr dφ = sin φ + e 2r sin φ 3e r + e r cos 2φ r π 2 = 0 5. A particle moves along the x-axis so that its velocity is proportional to the product of its instantaneous position x (measured from x = 0) and the time t (measured from t = 0). If the particle is located at x = 54 when t = 0 and x = 36 when t = 1, where will it be when t = 2? 6. Show that the differential equation dy dx = 4y 2 − x 4 4xy is non-separable but becomes separable on changing the dependent variable from y to v = y/x. Use this change of variable to find the solution to the original equation. 7. Solve (2y 2 + 4x 2 y) dx + (4xy + 3x 3) dy = 0 given that there exists an integrating factor of the form x p y q where p and q are constants .