Question

Elementary Differential Equations Problems:

1) Find the solution of the initial value problem of y" + 3y' = 0, y(0) = -2, y'(0) = 3

2) Find the general solution of the equation 4y" - 9y = 0

3) Find the general solution of the equation dy/dt = 2t(y – 2y²)

4) Given the second order linear homogeneous equation y"- 2y' + y = 0,

a) Verify that y₁(t) = e^t and y₂(t) = t e^t are solutions of the equation

b) Compute the Wronskian of y₁, y₂

c) Would y(t) = C₁y₁(t) + C₂y₂(t) also be a solution to the equation for arbitrary constants C₁,C₂? Explain.

5) Consider the second order non-linear differential equation (y * y") + (y’)² = 0 for t > 0

a) Verify that y₁(t) = 1 and y₂(t) = t^1/2 are solutions of this equation

b) Let y(t) = C₁y₁(t) + C₂y₂(t) = C₁+ (C₂ * t^1/2), for constants C₁,C₂. Compute y' and y"

c) Use the results from above to show that y(t) is NOT always a solution of the equation (y * y") + (y’)² = 0 for all C₁, C₂

Answer 1