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Chapter 2: Problem 6

Find the general solution of the given equation by the method given. a. \(y^{\prime \prime}-3 y^{\prime}+2 y=10\). Method of Undetermined Coefficients. b. \(y^{\prime \prime}+y^{\prime}=3 x^{2}\). Variation of Parameters.

Short Answer

Expert verified
For part a, the general solution is \(y = c_{1} e^{x} + c_{2} e^{2x} + 5\). For part b, the general solution is \(y = c_{1} + c_{2} e^{-x} - x^2 e^{x} + 2x e^{x} - 2 e^{x}\).

Step by step solution

01

Part a: Find Homogeneous Solution

First, find the homogeneous solution \(y_{h}\) of the equation \(y^{\prime \prime}-3 y^{\prime}+2 y=0\). This is done by solving its characteristic equation \(r^2 - 3r + 2 = 0\) yielding \(r = 1, 2\) and hence \(y_{h} = c_{1} e^{x} + c_{2} e^{2x}\).
02

Part a: Find Particular Solution

Next step is to find the particular solution \(y_{p}\) using the method of Undetermined Coefficients. By inspection of the RHS of the given equation, we select \(y_p = A\) as our guess since RHS is a constant. Substituting \(y_p = A\) into the given equation gives \(A = 5\), so \(y_p = 5\).
03

Part a: Combine Solutions to Find General Solution

The general solution is the sum of the homogeneous solution and the particular solution, hence \(y = y_h + y_p = c_{1} e^{x} + c_{2} e^{2x} + 5\).
04

Part b: Find Homogeneous Solution

First, find the homogeneous solution of the equation \(y^{\prime \prime}+y^{\prime}= 0\). The characteristic equation for the homogeneous equation \(r^{2} + r = 0\) yields \(r = -1, 0\), which gives \(y_{h} = c_{1} + c_{2} e^{-x}\).
05

Part b: Use Variation of Parameters to Find Particular Solution

Use the method of variation of parameters to find the particular solution \(y_p\). The Wronskian of the homogeneous equations is given by \(W = -e^{-x}\) and the particular solution will be \(y_p = - \int x^2 e^x dx = -x^2 e^{x} + 2x e^{x} - 2 e^{x}\).
06

Part b: Combine Solutions to Find General Solution

Combine the homogeneous solution and the particular solution to get the general solution, hence \(y = y_h + y_p = c_{1} + c_{2} e^{-x} - x^2 e^{x} + 2x e^{x} - 2 e^{x}\).

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Most popular questions from this chapter

Find all the solutions of the first-order differential equations. When an initial condition is given, find the particular solution satisfying that condition. a. \(\frac{d y}{d x}=\frac{e^{x}}{2 y}\). b. \(\frac{d y}{d t}=y^{2}\left(1+t^{2}\right), y(0)=1\). c. \(\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{x}\) d. \(x y^{\prime}=y(1-2 y), \quad y(1)=2\) e. \(y^{\prime}-(\sin x) y=\sin x\) f. \(x y^{\prime}-2 y=x^{2}, y(1)=1\) g. \(\frac{d s}{d t}+2 s=s t^{2}, \quad s(0)=1\) h. \(x^{\prime}-2 x=t e^{2 t}\) i. \(\frac{d y}{d x}+y=\sin x, y(0)=0\). j. \(\frac{d y}{d x}-\frac{3}{x} y=x^{3}, y(1)=4\)

A ball is thrown upward with an initial velocity of \(49 \mathrm{~m} / \mathrm{s}\) from \(539 \mathrm{~m}\) high. How high does the ball get, and how long does in take before it hits the ground? [Use results from the simple free fall problem, \(y^{\prime \prime}=-g\).]

Consider the flight of a golf ball with mass \(46 \mathrm{~g}\) and a diameter of \(42.7\) \(\mathrm{mm} .\) Assume it is projected at \(30^{\circ}\) with a speed of \(36 \mathrm{~m} / \mathrm{s}\) and no spin. a. Ignoring air resistance, analytically find the path of the ball and determine the range, maximum height, and time of flight for it to land at the height that the ball had started. b. Now consider a drag force \(f_{D}=\frac{1}{2} C_{D} \rho \pi r^{2} v^{2}\), with \(C_{D}=0.42\) and \(\rho=1.21 \mathrm{~kg} / \mathrm{m}^{3}\). Determine the range, maximum height, and time of flight for the ball to land at the height that it had started. c. Plot the Reynolds number as a function of time. [Take the kinematic viscosity of air, \(v=1.47 \times 10^{-5}\).] d. Based on the plot in part \(c\), create a model to incorporate the change in Reynolds number and repeat part b. Compare the results from parts a, b, and d.

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

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. $$ \begin{aligned} &x^{\prime}=2 x+3 y \\ &y^{\prime}=-3 x+2 y \end{aligned} $$
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