Chapter 3

In Problems \(3-8\), solve the given differential equation by using the substitution \(u=y^{\prime}\). $$ y^{2} y^{\prime \prime}=y^{\prime} $$

In Problems 1-18, solve the given differential equation. $$ x^{2} y^{\prime \prime}+3 x y^{\prime}-4 y=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}-y=\sinh 2 x $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ 4 y^{\prime \prime}-4 y^{\prime}-3 y=\cos 2 x $$

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ 6 y^{\prime \prime}+y^{\prime}-y=0 ; \quad y_{1}=e^{x / 3} $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &D x+D^{2} y=e^{3 t} \\ &(D+1) x+(D-1) y=4 e^{3 t} \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-y^{\prime}=-3\)

In September 1989, Hurricane Hugo hammered the coast of South Carolina with winds estimated at times to be as high as \(60.4 \mathrm{~m} / \mathrm{s}(135 \mathrm{mi} / \mathrm{h})\). Of the billions of dollars in damage, approximately \(\$ 420\) million of this was due to the market value of loblolly pine (Pinus taeda) lumber in the Francis Marion National Forest. One image from that storm remains hauntingly bizarre: all through the forest and surrounding region, thousands upon thousands of pine trees lay pointing exactly in the same direction, and all the trees were broken \(5-8\) meters from their base. In September 1996 , Hurricane Fran destroyed over \(8.2\) million acres of timber forest in eastern North Carolina. As happened seven years earlier, the planted loblolly trees all broke at approximately the same height. This seems to be a reproducible phenomenon, brought on by the fact that the trees in these planted forests are approximately the same age and size. In this problem, we are going to examine a mathematical model for the bending of loblolly pines in strong winds, and then use the model to predict the height at which a tree will break in hurricane-force winds. Wind hitting the branches of a tree transmits a force to the trunk of the tree. The runk is approximately a big cylindrical beam of length \(L\), and so we will model the deflection \(y(x)\) of the tree with the static beam equation \(E I^{(4)}=w(x)\) (equation (4) in this section), where \(x\) is distance measured in meters from ground level. Since the tree is rooted into the ground, the accompanying boundary conditions are those of a cantilevered beam: \(y(0)=0, y^{\prime}(0)=0\) at the rootedend, and \(y^{\prime \prime}(L)=0, y^{\prime \prime \prime}(L)=0\) at the free end, which is the top of the tree. (a) Loblolly pines in the foresthave the majority of theircrown (that is, branches and needles) in the upper \(50 \%\) of their length, so let's ignore the force of the wind on the lower portion of the tree. Furthermore, let's assume that the wind hitting the tree's crown results in a uniform load per unit length \(w_{0}\). In other words, the load on the tree is modeled by $$ w(x)=\left\\{\begin{array}{ll} 0, & 0 \leq x

Consider the initial-value problem $$ y^{\prime \prime}+y y^{\prime}=0, \quad y(0)=1, \quad y^{\prime}(0)=-1 $$ (a) Use the \(\mathrm{DE}\) and a numerical solver to graph the solution curve. (b) Find an explicit solution of the IVP. Use a graphing utility to graph this solution. (c) Find an interval of definition for the solution in part (b).

Solve the given differential equation. $$ 25 x^{2} y^{\prime \prime}+25 x y^{\prime}+y=0 $$

Find an interval centered about \(x=0\) for which the given initial-value problem has a unique solution. $$ (x-2) y^{\prime \prime}+3 y=x, y(0)=0, y^{\prime}(0)=1 $$

Give an interval over which \(f_{1}(x)=x^{2}\) and \(f_{2}(x)=x|x|\) are linearly independent. Then give an interval on which \(f_{1}\) and \(f_{2}\) are linearly dependent.

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &D x+D^{2} y=e^{3 t} \\ &(D+1) x+(D-1) y=4 e^{3 t} \end{aligned} $$

Blowing in the Wind In September 1989, Hurricane Hugo hammered the coast of South Carolina with winds estimated at times to be as high as \(60.4 \mathrm{~m} / \mathrm{s}(135 \mathrm{mi} / \mathrm{h})\). Of the billions of dollars in damage, approximately \(\$ 420\) million of this was due to the market value of loblolly pine (Pinus taeda) lumber in the Francis Marion National Forest. One image from that storm remains hauntingly bizarre: all through the forest and surrounding region, thousands upon thousands of pine trees lay pointing exactly in the same direction, and all the trees were broken 5-8 meters from their base. In September 1996, Hurricane Fran destroyed over \(8.2\) million acres of timber forest in eastern North Carolina. As happened seven years earlier, the planted loblolly trees all broke at approximately the same height. This seems to be a reproducible phenomenon, brought on by the fact that the trees in these planted forests are approximately the same age and size. In this problem, we are going to examine a mathematical model for the bending of loblolly pines in strong winds, and then use the model to predict the height at which a tree will break in hurricane-force winds.* Wind hitting the branches of a tree transmits a force to the munk of the tree. The trunk is approximately a big cylindrical beam of length \(L\), and so we will model the deflection \(y(x)\) of the tree with the static beam equation \(E I y^{(4)}=w(x)\) (equation (4) in this section), where \(x\) is distance measured in meters from ground level. Since the tree is rooted into the ground, the accompanying boundary conditions are those of a cantilevered beam: \(y(0)=0, y^{\prime}(0)=0\) at the rooted end, and \(y^{\prime \prime}(L)=0, y^{\prime \prime \prime}(L)=0\) at the free end, which is the top of the tree. (a) Loblolly pines in the forest have the majority of their crown (that is, branches and needles) in the upper \(50 \%\) of their length, so let's ignore the force of the wind on the lower portion of the tree. Furthermore, let's assume that the wind hitting the tree's crown results in a uniform load per unit length \(w_{0}\). In other words, the load on the tree is modeled by $$ w(x)= \begin{cases}0, & 0 \leq x

In Problems 1-18, solve the given differential equation. $$ 25 x^{2} y^{\prime \prime}+25 x y^{\prime}+y=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}-4 y=\frac{e^{2 x}}{x} $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}-y^{\prime}=-3 $$

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ x^{2} y^{\prime \prime}-7 x y^{\prime}+16 y=0 ; \quad y_{1}=x^{4} $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &D^{2} x-D y=t \\ &(D+3) x+(D+3) y=2 \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}+2 y^{\prime}=2 x+5-e^{-2 x}\)

Find two solutions of the initial-value problem $$ \left(y^{\prime \prime}\right)^{2}+\left(y^{\prime}\right)^{2}=1, \quad y(\pi / 2)=\frac{1}{2}, \quad y^{\prime}(\pi / 2)=\sqrt{3} / 2 $$ Use a numerical solver to graph the solution curves.

Solve the given differential equation. $$ 4 x^{2} y^{\prime \prime}+4 x y^{\prime}-y=0 $$

Solve each differential equation by variation of parameters. $$ y^{\prime \prime}-9 y=\frac{9 x}{e^{3 x}} $$

Find an interval centered about \(x=0\) for which the given initial-value problem has a unique solution. $$ y^{\prime \prime}+(\tan x) y=e^{x}, y(0)=1, y^{\prime}(0)=0 $$

Without the aid of the Wronskian determine whether the given set of functions is linearly independent or linearly dependent on the indicated interval. (a) \(f_{1}(x)=\ln x, f_{2}(x)=\ln x^{2},(0, \infty)\) (b) \(f_{1}(x)=x^{n}, f_{2}(x)=x^{n+1}, n=1,2, \ldots,(-\infty, \infty)\) (c) \(f_{1}(x)=x, f_{2}(x)=x+1,(-\infty, \infty)\) (d) \(f_{1}(x)=\cos (x+\pi / 2), f_{2}(x)=\sin x,(-\infty, \infty)\) (e) \(f_{1}(x)=0, f_{2}(x)=x,(-5,5)\) (f) \(f_{1}(x)=2, f_{2}(x)=2 x,(-\infty, \infty)\) (g) \(f_{1}(x)=x^{2}, f_{2}(x)=1-x^{2}, f_{3}(x)=2+x^{2},(-\infty, \infty)\) (h) \(f_{1}(x)=x e^{x+1}, f_{2}(x)=(4 x-5) e^{x}, f_{3}(x)=x e^{x},(-\infty, \infty)\)

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &D^{2} x-D y=t \\ &(D+3) x+(D+3) y=2 \end{aligned} $$

In Problems 1-18, solve the given differential equation. $$ 4 x^{2} y^{\prime \prime}+4 x y^{\prime}-y=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}-9 y=\frac{9 x}{e^{3 x}} $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}+2 y^{\prime}=2 x+5-e^{-2 x} $$

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ x^{2} y^{\prime \prime}+2 x y^{\prime}-6 y=0 ; \quad y_{1}=x^{2} $$

In Problems 9 and 10, find an interval centered about \(x=0\) for which the given initial-value problem has a unique solution. $$ y^{\prime \prime}+(\tan x) y=e^{x}, y(0)=1, y^{\prime}(0)=0 $$

The model \(m x^{\prime \prime}+k x+k_{1} x^{3}=F_{0} \cos \omega t\) of an undamped periodically driven spring/mass system is called Duffing's differential equation. Consider the initial-value problem \(x^{\prime \prime}+x+k_{1} x^{3}=5 \cos t, x(0)=1, x^{\prime}(0)=0 .\) Use a numerical solver to investigate the behavior of the system for values of \(k_{1}>0\) ranging from \(k_{1}=0.01\) to \(k_{1}=100\). State your conclusions.

In Problems, find the eigenvalues and eigenfunctions for the given boundary- value problem. $$ y^{\prime \prime}+\lambda y=0, y(0)=0, y(\pi)=0 $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\left(D^{2}-1\right) x-y=0 \\ &(D-1) x+D y=0 \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-y^{\prime}+\frac{1}{4} y=3+e^{x / 2}\)

In Problems, show that the substitution \(u=y^{\prime}\) leads to a Bernoulli equation. Solve this equation (see Section 2.5). $$ x y^{\prime \prime}=y^{\prime}+\left(y^{\prime}\right)^{3} $$

A mass weighing 64 pounds stretches a spring \(0.32\) foot. The mass is initially released from a point 8 inches above the equilibrium position with a downward velocity of \(5 \mathrm{ft} / \mathrm{s}\). (a) Find the equation of motion. (b) What are the amplitude and period of motion? (c) How many complete cycles will the mass have completed at the end of \(3 \pi\) seconds? (d) At what time does the mass pass through the equilibrium position beading downward for the second time? (e) At what time does the mass attain its extreme displacement on either side of the equilibrium position? (f) What is the position of the mass at \(t=3\) s? (g) What is the instantaneous velocity at \(t=3 \mathrm{~s}\) ? (h) What is the acceleration at \(t=3 \mathrm{~s} ?\) (i) What is the instantaneous velocity at the times when the mass passes through the equilibrium position? (j) At what times is the mass 5 inches below the equilibrium position? (k) At what times is the mass 5 inches below the equilibrium position heading in the upward direction?

Solve the given differential equation. $$ x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0 $$

Solve each differential equation by variation of parameters. $$ y^{\prime \prime}+3 y^{\prime}+2 y=\frac{1}{1+e^{x}} $$

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\left(D^{2}-1\right) x-y=0 \\ &(D-1) x+D y=0 \end{aligned} $$

In Problems 11-20, find the eigenvalues and eigenfunctions for the given boundary-value problem. $$ y^{\prime \prime}+\lambda y=0, y(0)=0, y(\pi)=0 $$

In Problems 1-18, solve the given differential equation. $$ x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}+3 y^{\prime}+2 y=\frac{1}{1+e^{x}} $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}-y^{\prime}+\frac{1}{4} y=3+e^{x / 2} $$

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ x y^{\prime \prime}+y^{\prime}=0 ; \quad y_{1}=\ln x $$

In Problems, find the eigenvalues and eigenfunctions for the given boundary- value problem. $$ y^{\prime \prime}+\lambda y=0, y^{\prime}(0)=0, y(\pi / 4)=0 $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\left(2 D^{2}-D-1\right) x-(2 D+1) y=1 \\ &(D-1) x+D y=-1 \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-16 y=2 e^{4 x}\)

In Problems , show that the substitution \(u=y^{\prime}\) leads to a Bernoulli equation. Solve this equation (see Section 2.5). $$ x y^{\prime \prime}=y^{\prime}+x\left(y^{\prime}\right)^{2} $$

Proceed as in Example 2 to find the general solution of the given differential equation. Use the results obtained in Problems \(1-6 .\) Do not evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}-2 y^{\prime}+2 y=\cos ^{2} x\)