1.4: Problems

1.1. Find all of the solutions of the first order differential equations. When an initial condition is given, find the particular solution satisfying that condition.

a. \(\dfrac{d y}{d x}=\dfrac{\sqrt{1-y^{2}}}{x}\)

b. \(x y^{\prime}=y(1-2 y), \quad y(1)=2\)

c. \(y^{\prime}-(\sin x) y=\sin x\)

d. \(x y^{\prime}-2 y=x^{2}, y(1)=1\)

e. \(\dfrac{d s}{d t}+2 s=s t^{2}, \quad, s(0)=1\)

f. \(x^{\prime}-2 x=t e^{2 t}\)

1.2. Find all of the solutions of the second order differential equations. When an initial condition is given, find the particular solution satisfying that condition.

a. \(y^{\prime \prime}-9 y^{\prime}+20 y=0\)

b. \(y^{\prime \prime}-3 y^{\prime}+4 y=0, \quad y(0)=0, \quad y^{\prime}(0)=1\)

c. \(x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0, \quad x>0\)

d. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+3 y=0, \quad x>0\)

1.3. Consider the differential equation

\[\dfrac{d y}{d x}=\dfrac{x}{y}-\dfrac{x}{1+y} \nonumber \]

a. Find the 1-parameter family of solutions (general solution) of this equation.

b. Find the solution of this equation satisfying the initial condition \(y(0)=1\). Is this a member of the 1-parameter family?

1.4. The initial value problem

\[\dfrac{d y}{d x}=\dfrac{y^{2}+x y}{x^{2}}, \quad y(1)=1 \nonumber \]

does not fall into the class of problems considered in our review. However, if one substitutes \(y(x)=x z(x)\) into the differential equation, one obtains an equation for \(z(x)\) which can be solved. Use this substitution to solve the initial value problem for \(y(x)\).

1.5. Consider the nonhomogeneous differential equation \(x^{\prime \prime}-3 x^{\prime}+2 x=6 e^{3 t}\).

a. Find the general solution of the homogenous equation.

b. Find a particular solution using the Method of Undetermined Coefficients by guessing \(x_{p}(t)=A e^{3 t}\)

c. Use your answers in the previous parts to write down the general solution for this problem.

1.6. Find the general solution of each differential equation. When an initial condition is given, find the particular solution satisfying that condition.

a. \(y^{\prime \prime}-3 y^{\prime}+2 y=20 e^{-2 x}, \quad y(0)=0, \quad y^{\prime}(0)=6\)

b. \(y^{\prime \prime}+y=2 \sin 3 x\)

c. \(y^{\prime \prime}+y=1+2 \cos x\)

d. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=3 x^{2}-x, \quad x>0\)

1.7. Verify that the given function is a solution and use Reduction of Order to find a second linearly independent solution.

a. \(x^{2} y^{\prime \prime}-2 x y^{\prime}-4 y=0, \quad y_{1}(x)=x^{4}\)

b. \(x y^{\prime \prime}-y^{\prime}+4 x^{3} y=0, \quad y_{1}(x)=\sin \left(x^{2}\right)\)

1.8. A certain model of the motion of a tossed whiffle ball is given by

\[m x^{\prime \prime}+c x^{\prime}+m g=0, \quad x(0)=0, \quad x^{\prime}(0)=v_{0} \nonumber \]

Here \(m\) is the mass of the ball, \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) is the acceleration due to gravity and \(c\) is a measure of the damping. Since there is no \(x\) term, we can write this as a first order equation for the velocity \(v(t)=x^{\prime}(t)\):

\[m v^{\prime}+c v+m g=0 \nonumber \]

a. Find the general solution for the velocity \(v(t)\) of the linear first order differential equation above.

b. Use the solution of part a to find the general solution for the position \(x(t)\).

c. Find an expression to determine how long it takes for the ball to reach it's maximum height?

d. Assume that \(c / m=10 \mathrm{~s}^{-1}\). For \(v_{0}=5,10,15,20 \mathrm{~m} / \mathrm{s}\), plot the solution, \(x(t)\), versus the time.

e. From your plots and the expression in part c, determine the rise time. Do these answers agree?

f. What can you say about the time it takes for the ball to fall as compared to the rise time?