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Chapter 2: Problem 2

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

Short Answer

Expert verified
a) \(y = c_1 e^{4x} + c_2 e^{5x}\). b) \(y = e^{1.5x}( A\cos(\sqrt{7/4}x) + B\sin(\sqrt{7/4}x))\), with A and B are determined by initial conditions. c) \(y = c_1 x^{m_1} + c_2 x^{m_2}\), where \(m_1\) and \(m_2\) are roots of the characteristic equation. d) Similar to c, with different roots.

Step by step solution

01

Solving equation a

To find the solution of the second-order differential equation \(y^{\prime \prime}-9 y^{\prime}+20 y=0\), we start by solving the characteristic equation \(m^2 - 9m + 20 = 0\). Solving this quadratic equation gives the roots \(m=4\) and \(m=5\). Therefore, the general solution of the second-order differential equation is \(y=c_1 e^{4x} + c_2 e^{5x}\) where \(c_1\) and \(c_2\) are arbitrary constants.
02

Solving equation b

Start by solving the given initial value problem's characteristic equation, \(y^{\prime \prime}-3 y^{\prime}+4 y=0 \). The characteristic equation is \(m^2 - 3m + 4 = 0\), which has roots \(m=1.5 \pm \sqrt{7/4}\). The two solutions are complex and conjugate, so the general solution is in the form \((y= e^{mt}( A\cos(nt) + B\sin(nt)))\). Here, \(m\) is the real part and \(n\) is the imaginary part of the roots. To solve for particular solution that satisfies \(y(0)=0\) and \(y^{\prime}(0)=1\), plug those conditions into your general solution and its derivative.
03

Solving equations c and d

These are examples of the Cauchy-Euler type of differential equations. Equation c, \(x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0\) has the general solution \(y = c_1 x^{m_1} + c_2 x^{m_2}\) where \(m_1\) and \(m_2\) are roots of the characteristic equation \(m(m-1) + 5m + 4 = 0\). Similarly, equation d, \(x^{2} y^{\prime \prime}-2 x y^{\prime}+3 y=0\), has the same form, with roots of \(m(m-1) - 2m + 3 = 0\).

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Most popular questions from this chapter

Consider the flight of a tennis ball with mass \(57 \mathrm{~g}\) and a diameter of \(66.0 \mathrm{~mm}\). Assume the ball is served \(6.40\) meters from the net at a speed of \(50.0 \mathrm{~m} / \mathrm{s}\) down the center line from a height of \(2.8 \mathrm{~m}\). It needs to just clear the net \((0.914 \mathrm{~m})\). a. Ignoring air resistance and spin, analytically find the path of the ball assuming it just clears the net. Determine the angle to clear the net and the time of flight. b. Find the angle to clear the net assuming the tennis ball is given a topspin with \(w=50 \mathrm{rad} / \mathrm{s}\) c. Repeat part b assuming the tennis ball is given a bottom spin with \(\omega=50 \mathrm{rad} / \mathrm{s}\) d. Repeat parts \(\mathrm{a}, \mathrm{b}\), and \(\mathrm{c}\) with a drag force, taking \(C_{D}=0.55\).

A spring fixed at its upper end is stretched 6 inches by a 10-pound weight attached at its lower end. The spring-mass system is suspended in a viscous medium so that the system is subjected to a damping force of \(5 \frac{d x}{d t}\) lbs. Describe the motion of the system if the weight is drawn down an additional 4 inches and released. What would happen if you changed the coefficient " 5 " to "4"? [You may need to consult your introductory physics text.]

Consider the following systems. Determine the families of orbits for each system and sketch several orbits in the phase plane and classify them by their type (stable node, etc.). \(\mathrm{a}\). $$ \begin{aligned} &x^{\prime}=3 x \\ &y^{\prime}=-2 y \end{aligned} $$ b. $$ \begin{aligned} &x^{\prime}=-y \\ &y^{\prime}=-5 x \end{aligned} $$ c. $$ \begin{aligned} &x^{\prime}=2 y \\ &y^{\prime}=-3 x \end{aligned} $$ d. $$ \begin{aligned} &x^{\prime}=x-y \\ &y^{\prime}=y \end{aligned} $$ e. $$ \begin{aligned} &x^{\prime}=2 x+3 y \\ &y^{\prime}=-3 x+2 y \end{aligned} $$

For the problem \(y^{\prime \prime}-k^{2} y=f(x), y(0)=0, y^{\prime}(0)=1\), a. Find the initial value Green's function. b. Use the Green's function to solve \(y^{\prime \prime}-y=e^{-x}\). c. Use the Green's function to solve \(y^{\prime \prime}-4 y=e^{2 x}\).

Consider the system $$ \begin{array}{r} x^{\prime}=-4 x-y \\ y^{\prime}=x-2 y \end{array} $$ a. Determine the second-order differential equation satisfied by \(x(t)\). b. Solve the differential equation for \(x(t)\). c. Using this solution, find \(y(t)\). d. Verify your solutions for \(x(t)\) and \(y(t)\). e. Find a particular solution to the system given the initial conditions \(x(0)=1\) and \(y(0)=0\)
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