Chapter 3

In Problems \(41-44\), use systematic elimination to solve the given system. $$ \begin{aligned} (D+2) x+(D+1) y &=\sin 2 t \\ 5 x+(D+3) y &=\cos 2 t \end{aligned} $$

Consider a driven undamped spring/mass system described by the initial-value problem $$ \frac{d^{2} x}{d t^{2}}+\omega^{2} x=F_{0} \sin ^{n} \gamma t, x(0)=0, x^{\prime}(0)=0 $$ (a) For \(n=2\), discuss why there is a single frequency \(\gamma_{1} / 2 \pi\) at which the system is in pure resonance. (b) For \(n=3\), discuss why there are two frequencies \(\gamma_{1} / 2 \pi\) and \(\gamma_{2} / 2 \pi\) at which the system is in pure resonance. (c) Suppose \(\omega=1\) and \(F_{0}=1\). Use a numerical solver to obtain the graph of the solution of the initial-value problem for \(n=2\) and \(\gamma=\gamma_{1}\) in part (a). Obtain the graph of the solution of the initial-value problem for \(n=3\) corresponding, in turn, to \(\gamma=\gamma_{1}\) and \(\gamma=\gamma_{2}\) in part (b).

In Problems 43-48, use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 3.3-3.5. $$ x^{2} y^{\prime \prime}-9 x y^{\prime}+25 y=0 $$

Find the charge on the capacitor in an \(L R C\) -series circuit at \(t=0.01 \mathrm{~s}\) when \(L=0.05 \mathrm{~h}, R=2 \Omega, C=0.01 \mathrm{f}, E(t)=0 \mathrm{~V}\) \(q(0)=5 \mathrm{C}\), and \(i(0)=0\) A. Determine the first time at which the charge on the capacitor is equal to zero.

A free undamped spring/mass system oscillates with a period of \(3 \mathrm{~s}\). When \(8 \mathrm{lb}\) is removed from the spring, the system then has a period of \(2 \mathrm{~s}\). What was the weight of the original mass on the spring?

Suppose the solution of the boundary-value problem $$ y^{\prime \prime}+P y^{\prime}+Q y \quad f(x), \quad y(a) \quad 0, y(b) \quad 0 $$ \(a

In Problems 43-48, use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 3.3-3.5. $$ x^{2} y^{\prime \prime}+10 x y^{\prime}+8 y=x^{2} $$

In Problems 46 and 47, find a particular solution of the given differential equation. Use a CAS as an aid in carrying out differentiations, simplifications, and algebra. \(y^{\prime \prime}-4 y^{\prime}+8 y=\left(2 x^{2}-3 x\right) e^{2 x} \cos 2 x+\left(10 x^{2}-x-1\right) e^{2 x} \sin 2 x\)

Use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 3.3-3.5. $$ x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=\ln x^{2} $$

Find the charge on the capacitor in an \(L R C\) -series circuit when \(L=\frac{1}{4} \mathrm{~h}, R=20 \Omega, C=\frac{1}{300} \mathrm{f}, E(t)=0 \mathrm{~V}, q(0)=4 \mathrm{C}\), and \(i(0)=0 \mathrm{~A}\). Is the charge on the capacitor everequal to zero?

In Problems 43-48, use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 3.3-3.5. $$ x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=\ln x^{2} $$

Find a particular solution of the given differential equation. Use a CAS as an aid in carrying out differentiations, simplifications, and algebra. \(y^{(4)}+2 y^{\prime \prime}+y=2 \cos x-3 x \sin x\)

In Problems 43-48, use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections 3.3-3.5. $$ x^{2} y^{\prime \prime}-3 x y^{\prime}+13 y=4+3 x $$

In Problems 46 and 47, find a particular solution of the given differential equation. Use a CAS as an aid in carrying out differentiations, simplifications, and algebra. $$ y^{(4)}+2 y^{\prime \prime}+y=2 \cos x-3 x \sin x $$

Find the charge on the capacitor and the current in the given \(L R C\) -series circuit. Find the maximum charge on the capacitor. $$ \begin{aligned} &L=1 \mathrm{~h}, R=100 \Omega, C=0.0004 \mathrm{f}, E(t)=30 \mathrm{~V}, q(0)=0 \mathrm{C}, \\ &i(0)=2 \mathrm{~A} \end{aligned} $$

A32-pound weight stretchesa spring 6 inches. The weight moves through a medium offering a damping force numerically equal to \(\beta\) times the instantaneous velocity. Determine the values of \(\beta\) for which the system will exhibit oscillatory motion.

In Problems 47 and 48 , find the charge on the capacitor and the current in the given \(L R C\)-series circuit. Find the maximum charge on the capacitor. $$ \begin{aligned} &L=1 \mathrm{~h}, R=100 \Omega, C=0.0004 \mathrm{f}, E(t)=30 \mathrm{~V}, q(0)=0 \mathrm{C} \\ &i(0)=2 \mathrm{~A} \end{aligned} $$

Use the substitution \(t=-x\) to solve the given initial-value problem on the interval \((-\infty, 0)\). $$ 4 x^{2} y^{\prime \prime}+y=0, y(-1)=2, y^{\prime}(-1)=4 $$

Find the steady-state charge and the steady-state current in an \(L R C\) -series circuit when \(L=1 \mathrm{~h}, R=2 \Omega, C=0.25 \mathrm{f}\), and \(E(t)=50 \cos t \mathrm{~V}\)

A series circuit contains an inductance of \(L=1 \mathrm{~h}\), a capacitance of \(C=10^{-4} f\), and an electromotive force of \(E(t)=\) \(100 \sin 50 t \mathrm{~V}\). Initially the charge \(q\) and current \(i\) are zero. (a) Find the equation for the charge at time \(t\). (b) Find the equation for the current at time \(t\). (c) Find the times for which the charge on the capacitor is zero.

In Problems 49 and 50, use the substitution \(t=-x\) to solve the given initial- value problem on the interval \((-\infty, 0)\). $$ 4 x^{2} y^{\prime \prime}+y=0, y(-1)=2, y^{\prime}(-1)=4 $$

In Problems 49-58 find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} e^{x}+c_{2} e^{6 x} $$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} e^{-5 x}+c_{2} e^{-4 x} $$

Use the substitution \(t=-x\) to solve the given initial-value problem on the interval \((-\infty, 0)\). $$ x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0, y(-2)=8, y^{\prime}(-2)=0 $$

In Problems 49 and 50, use the substitution \(t=-x\) to solve the given initial- value problem on the interval \((-\infty, 0)\). $$ x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0, y(-2)=8, y^{\prime}(-2)=0 $$

In Problems 49-58 find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} e^{-5 x}+c_{2} e^{-4 x} $$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1}+c_{2} e^{3 x} $$

In Problems 49-58 find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1}+c_{2} e^{3 x} $$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} e^{-10 x}+c_{2} x e^{-10 x} $$

Find a Cauchy-Euler differential equation of lowest order with real coefficients if it is known that 2 and \(1-i\) are two roots of its auxiliary equation.

Find the steady-state current in an \(L R C\) -series circuit when \(L=\frac{1}{2} \mathrm{~h}, R=20 \Omega, C=0.001 \mathrm{f}\), and \(E(t)=100 \sin 60 t+\) \(200 \cos 40 t \mathrm{~V}\)

In Problems 49-58 find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} e^{-10 x}+c_{2} x e^{-10 x} $$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} \cos 8 x+c_{2} \sin 8 x $$

The initial conditions \(y(0)=y_{0}, y^{\prime}(0)=y_{1}\), apply to each of the following differential equations: $$ \begin{aligned} &x^{2} y^{\prime \prime}=0 \\ &x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=0 \\ &x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0 \end{aligned} $$ For what values of \(y_{0}\) and \(y_{1}\) does each initial-value problem have a solution?

Find the charge on the capacitor in an \(L R C\) -series circuit when \(L=\frac{1}{2} \mathrm{~h}, R=10 \Omega, C=0.01 \mathrm{f}, E(t)=150 \mathrm{~V}, q(0)=1 \mathrm{C}\), and \(i(0)=0 \mathrm{~A} .\) What is the charge on the capacitor after a long time?

In Problems 49-58 find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} \cos 8 x+c_{2} \sin 8 x $$

In Problems 49-58 find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} \cosh \frac{1}{2} x+c_{2} \sinh \frac{1}{2} x $$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} e^{x} \cos x+c_{2} e^{x} \sin x $$

In Problems 55-58, solve the given differential equation by using a CAS to find the (approximate) roots of the auxiliary equation. $$ 2 x^{3} y^{\prime \prime \prime}-10.98 x^{2} y^{\prime \prime}+8.5 x y^{\prime}+1.3 y=0 $$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1}+c_{2} e^{-2 x} \cos 5 x+c_{3} e^{-2 x} \sin 5 x $$

Find the charge on the capacitor and the current in an \(L C\)-circuit when \(L=0.1 \mathrm{~h}, C=0.1 \mathrm{f}, E(t)=100 \sin \gamma t \mathrm{~V}, q(0)=0 \mathrm{C}\), and \(i(0)=0 \mathrm{~A}\).

In Problems 55-58, solve the given differential equation by using a CAS to find the (approximate) roots of the auxiliary equation. $$ x^{3} y^{\prime \prime \prime}+4 x^{2} y^{\prime \prime}+5 x y^{\prime}-9 y=0 $$

In Problems 49-58 find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1}+c_{2} e^{-2 x} \cos 5 x+c_{3} e^{-2 x} \sin 5 x $$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1}+c_{2} x+c_{3} e^{7 x} $$

In Problems 55-58, solve the given differential equation by using a CAS to find the (approximate) roots of the auxiliary equation. $$ x^{4} y^{(4)}+6 x^{3} y^{\prime \prime \prime}+3 x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=0 $$

In Problems 49-58 find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1}+c_{2} x+c_{3} e^{7 x} $$

Find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} \cos x+c_{2} \sin x+c_{3} \cos 3 x+c_{4} \sin 3 x $$

In Problems 55-58, solve the given differential equation by using a CAS to find the (approximate) roots of the auxiliary equation. $$ x^{4} y^{(4)}-6 x^{3} y^{\prime \prime \prime}+33 x^{2} y^{\prime \prime}-105 x y^{\prime}+169 y=0 $$

In Problems 49-58 find a homogeneous linear differential equation with constant coefficients whose general solution is given. $$ y \quad c_{1} \cos x+c_{2} \sin x+c_{3} \cos 3 x+c_{4} \sin 3 x $$

The Paris Guns The first mathematically correct theory of projectile motion was originally formulated by Galileo Galilei (1564-1642), then clarified and extended by bis younger collaborators Bonaventara Cavalieri (1598-1647) and Evangelista Torricelli (1608-1647). Galileo's theory was based on two simple hypotheses suggested by experimental observations: that a projectile moves with constant horizontal velocity and with constant downward vertical acceleration. Galileo, Cavalieri, and Toricelli did not have calculus at their disposal, 80 their arguments were largely geometric, but we can reproduce their results using a system of differential equations. Suppose that a projectile is launched from ground level at an angle \(\theta\) with respect to the horizontal and with an initial velocity of magnitude \(\left\|v_{0}\right\|=v_{0} \mathrm{~m} / \mathrm{s}\). Let the projectile's height above the ground be \(y\) meters and its harizontal distance from the launch site be \(x\) meters, and for convenience take the launch site to be the origin in the \(x y\)-plane. Then Galileo's hypotheses can be represented by the following initial-value problem: $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}=0 \\ &\frac{d^{2} y}{d t^{2}}=-g \end{aligned} $$ where \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}, x(0)=0, y(0)=0, x^{\prime}(0)=v_{0} \cos \theta\) (the \(x\)-component of the initial velocity), and \(y^{\prime}(0)=v_{0} \sin \theta\) (the \(y\)-component of the initial velocity). See FICURE \(3 . R .4\) and Problem 23 in Exercises 3.12. FICURE 3.R.4 Ballistic projectile (a) Note that the system of equations in (1) is decoupled, that is, itconsists of separate differential equations for \(x(t)\) and \(y(t)\). Moreover, each of these differential equations can be solved simply by antidifferentiating twice. Solve (1) to obtain explicit formulas for \(x(t)\) and \(y(t)\) in terms of \(v_{0}\) and \(\theta\). Then algebraically eliminate \(t\) to show that the trajectory of the projectile in the \(x y\)-plane is a parabola. (b) A central question throughout the history of ballistics has been this: Given a gun that fires a projectile with a certain initial speed \(v_{0}\), at what angle with respect to the horizontal should the gunbe fired to maximize its range? The range is the horizontal distance traversed by a projectile before it hits the ground. Show that according to (1), the range of the projectile is \(\left(v_{0}^{2} / g\right) \sin 2 \theta\), so that a maximum range \(v_{0}^{2} / g\) is achieved for the launch angle \(\theta=\pi / 4=45^{\circ}\). (c) Show that the maximum height attained by the projectile if launched with \(\theta=45^{\circ}\) for maximum range is \(v_{0}^{2} /(4 \mathrm{~g})\).