Chapter 10

Solve the initial value problem \(y^{\prime \prime}+6 y^{\prime}+18 y=0, y(0)=0, y^{\prime}(0)=6\).

Find the general solution of the equation. $$y^{\prime}+y \sec t=\tan t,-\pi / 2

Solve the initial value problem. $$y^{\prime}+y \cos \left(e^{t}\right)=0, y(0)=0$$

Find a non-constant solution of the initial value problem \(y^{\prime}=y^{1 / 3}, y(0)=0,\) using separation of variables. Note that the constant function \(y(t)=0\) also solves the initial value problem. This shows that an initial value problem can have more than one solution.

Find the general solution to the differential equation. $$y^{\prime \prime}+12 y^{\prime}+36 y=6 e^{-6 t}$$

Solve the initial value problem \(y^{\prime \prime}+4 y=0, y(0)=\sqrt{3}, y^{\prime}(0)=2\).

Solve the initial value problem. $$t y^{\prime}-2 y=0, y(1)=4$$

Solve the equation for Newton's law of cooling leaving \(M\) and \(k\) unknown.

Find the general solution to the differential equation. $$y^{\prime \prime}-8 y^{\prime}+16 y=-2 e^{4 t}$$

Solve the initial value problem \(y^{\prime \prime}+100 y=0, y(0)=5, y^{\prime}(0)=50 .\)

Solve the initial value problem. $$t^{2} y^{\prime}+y=0, y(1)=-2, t>0$$

Find the general solution to the differential equation. $$y^{\prime \prime}+6 y^{\prime}+5 y=4$$

Solve the initial value problem \(y^{\prime \prime}+4 y^{\prime}+13 y=0, y(0)=1, y^{\prime}(0)=1\).

Solve the initial value problem. $$t^{3} y^{\prime}=2 y, y(1)=1, t>0$$

Solve the logistic equation \(y^{\prime}=k y(M-y) .\) (This is a somewhat more reasonable population model in most cases than the simpler \(\left.y^{\prime}=k y .\right)\) Sketch the graph of the solution to this equation when \(M=1000, k=0.002, y(0)=1\).

Find the general solution to the differential equation. $$y^{\prime \prime}+5 y=8 \sin (2 t)$$

Solve the initial value problem \(y^{\prime \prime}-8 y^{\prime}+25 y=0, y(0)=3, y^{\prime}(0)=0\).

Solve the initial value problem. $$t^{3} y^{\prime}=2 y, y(1)=0, t>0$$

Find the general solution to the differential equation. $$y^{\prime \prime}+5 y=8 \sin (2 t)$$

A function \(y(t)\) is a solution of \(y^{\prime}+k y=0 .\) Suppose that \(y(0)=100\) and \(y(2)=4 .\) Find \(k\) and find \(y(t)\).

A radioactive substance obeys the equation \(y^{\prime}=k y\) where \(k<0\) and \(y\) is the mass of the substance at time t. Suppose that initially, the mass of the substance is \(y(0)=M>0 .\) At what time does half of the mass remain? (This is known as the half life. Note that the half life depends on \(k\) but not on M.)

Find the general solution to the differential equation. $$y^{\prime \prime}-4 y=4 e^{2 t}$$

A function \(y(t)\) is a solution of \(y^{\prime}+t^{k} y=0 .\) Suppose that \(y(0)=1\) and \(y(1)=e^{-13}\). Find \(k\) and find \(y(t)\).

Bismuth- 210 has a half life of five days. If there is initially 600 milligrams, how much is left after 6 days? When will there be only 2 milligrams left?

Solve the initial value problem. $$y^{\prime \prime}-y=3 t+5, y(0)=0, y^{\prime}(0)=0$$

Consider the differential equation \(a y^{\prime \prime}+b y^{\prime}=0,\) with a and \(b\) both non-zero. Find the general solution by the method of this section. Now let \(g=y^{\prime} ;\) the equation may be written as \(a g^{\prime}+b g=0,\) a first order linear homogeneous equation. Solve this for \(g,\) then use the relationship \(g=y^{\prime}\) to find \(y\).

A bacterial culture grows at a rate proportional to its population. If the population is one million at \(t=0\) and 1.5 million at \(t=1\) hour, find the population as a function of time.

The half life of carbon-14 is 5730 years. If one starts with 100 milligrams of carbon-14, how much is left after 6000 years? How long do we have to wait before there is less than 2 milligrams?

Solve the initial value problem. $$y^{\prime \prime}+9 y=4 t, y(0)=0, y^{\prime}(0)=0$$

Suppose that \(y(t)\) is a solution to \(a y^{\prime \prime}+b y^{\prime}+c y=0, y\left(t_{0}\right)=0, y^{\prime}\left(t_{0}\right)=0 .\) Show that \(y(t)=0\).

A radioactive element decays with a half-life of 6 years. If a mass of the element weighs ten pounds at \(t=0,\) find the amount of the element at time \(t\).

Solve the initial value problem. $$y^{\prime \prime}+12 y^{\prime}+37 y=10 e^{-4 t}, y(0)=4, y^{\prime}(0)=0$$

Solve the initial value problem. $$y^{\prime \prime}+6 y^{\prime}+18 y=\cos t-\sin t, y(0)=0, y^{\prime}(0)=2$$

Find the solution for the mass-spring equation \(y^{\prime \prime}+4 y^{\prime}+29 y=689 \cos (2 t)\).

Find the solution for the mass-spring equation \(3 y^{\prime \prime}+12 y^{\prime}+24 y=2 \sin t .\)

Consider the differential equation \(m y^{\prime \prime}+b y^{\prime}+k y=\cos (\omega t),\) with \(m, b,\) and \(k\) all positive and \(b^{2}<2 m k ;\) this equation is a model for a damped mass-spring system with external driving force \(\cos (\omega t) .\) Show that the steady state part of the solution has amplitude $$\frac{1}{\sqrt{\left(k-m \omega^{2}\right)^{2}+\omega^{2} b^{2}}}$$ Show that this amplitude is largest when \(\omega=\frac{\sqrt{4 m k-2 b^{2}}}{2 m} .\) This is the resonant frequency of the system.