Differential Equations

1) Compute y' and y'' and then combine these derivatives with y as a linear second-order differential equation that is free of the symbols c₁ and c₂ and has the form F(y, y', y'')=0. The symbols c₁ and c₂ represent constants.

                                                                                    Y= c₁e^x + c₂xe^x

2) Verify that the indicated function is a particular solution of the given differential equation. Give an interval of definition I for each solution.

y'' + y = sexx;  y=xsinx + (cosx)ln(cosx)

3) Solve the differential equation (2x+y+1)y' =1

4) Solve the given initial-value problem and give the largest interval I on which the solution is defined.                                

dy/dt + 2(t+1)y² = 0, y(0) = -1/8

5) According to Stefan's law of radiation the absolute temperature T of a body cooling in a medium at constant absolute temperature T_m is given by

dT/dt = k(T⁴ - T⁴_m),  solve the differential equation.

6) Using the information provided in problem #5, show that when T-T_m is small in comparison to T_m then Newton's law of cooling approximates Stefan's law.

7) Find the general solution of the differential equation.  2y'' + 2y' +3y =0

8) Solve the differential equation subject to the indicated conditions.

Y'' + 2y' +y=0, y(-1) = 0, y'(0) = 0

9) A mass weighing 32 pounds stretches a spring 6 inches. The mass moves thought a medium offering a damping force that is numerically equal to ? times the instantaneous velocity. Determine the values of ? >0 for which the spring? Mass system will exhibit oscillatory motion.

10) The vertical motion of a mass attached to a spring is describe by the IVP 1/4x'' + x' + x = 0, x(0)=4, x'(0)=2. Determine the maximum vertical displacement of the mass.