# Questions in Section Differential Equations: page 7

Find the solution to the differential equation as a convolution integral: y''+9y=sin(t), y(0)=0, y'(0)=0.

Use reduction of order to find a second solution to differential equation x^2y''-6y=0, given that first solution is y_1=x^3.

A cup of coffee at 90°C is poured into a mug and left in a room at 21°C. After one minute, the coffee temperature is 85°C. Suppose that the coffee temperature does obey Newton’s Law of Cooling. The coffee becomes safe to drink after it cools to 60°C. How long will it take before you can drink the coffee, this means at which time is the coffee temperature 60°C?

Consider the IVP: y'=2x+y; y(0)=1.

1. Find the actual value of y(1).
2. Use Euler's method to approximate the value of y(1) using a step size of h=0.2.
3. Calculate the absolute error and the relative (percentage) error of your approximate value.

Consider the first-order differential equation ydx+(2x+e^y/y)dy=0.

Notice that it is not a separable, linear, homogeneous, Bernoulli or exact differential equation.

1. Obtain a partial differential equation for the integrating factor mu(x,y) for this DE to be exact.
2. Show that if we assume that mu is only a function of x then the DE does not make sense.
3. Show that if we assume that mu is only a function of y then the DE does make sense. Hence, solve this DE and show that mu=Cy.
4. By using integrating factor mu=y, show that the DE becomes an exact DE and hence show that the implicit solution to the DE is xy^2+e^y=C.

A mass weighing 32 lb stretches a spring 8 ft. There is no damping. There is a force of e^t. Find the general solution to the differential equation: m u''+gammau'+ku=F(t), u(0)=0, u'(0)=0.