Identify whether the differential equation is linear

Question

1. Determine whether the equation y - ln y = x² + 1 an implicit solution to the differential equation. Show your work.

dy/dx = 2xy/y-1

2. Identify whether the differential equation is linear, separable, neither, or both. Obtain the general solution y(x) using the appropriate method.

x(dy/dx) + 2y = x^-3

3. A cup of coffee initially at 180^o F cools to 150^o F after five minutes sitting in a room of constant temperature 70^o F. Determine when the coffee will cool to 100^o F.

4. Find a particular solution to the differential equation using the method of undetermined coefficients.

y'' + 4y' + 4y = 8t²

5. A 2-kg mass is attached to a spring with stiffness 20 N/m. The damping constant for the spring is 14 N-sec/m. If the mass is pulled 1 to the right of equilibrium and given an initial rightward velocity of 2 m/sec, when will the mass first cross the equilibrium position?

6. A 1-kg is attached to a spring with stiffness 10 N/m. The damping constant for the spring is 2 N-sec/m. If the mass is pulled 2 m to the left of equilibrium and given an initial rightward velocity of 2 m/sec, when will the mass first cross the equilibrium position?

7. Find a geranial solution

y'' + 4y' + 4y = 4t² + 2e^2t lnt;