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`.

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