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Chapter 20: Problem 1

Obtain the general solutions, that is the complementary functions, of the following homogeneous equations: (a) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 \frac{\mathrm{d} y}{\mathrm{~d} x}+y=0\) (b) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} t^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} t}+5 y=0\) (c) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} x}-2 y=0\) (d) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+9 y=0\) (e) \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 \frac{\mathrm{d} y}{\mathrm{~d} x}=0\) (f) \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-16 x=0\)

Short Answer

Expert verified
(a) \(y'' - 2y' + y = 0\) (b) \(y'' + y' + 5y = 0\) (c) \(y'' + y' - 2y = 0\) (d) \(y'' + 9y = 0\) (e) \(y'' - 2y' = 0\) (f) \(x'' - 16x = 0\)

Step by step solution

01

Identify coefficients

For this equation, we have the following coefficients: \(a = 1\), \(b = -2\), and \(c = 1\).
02

Form the characteristic equation

Our characteristic equation will be: \(r^2 - 2r + 1 = 0\).
03

Solve the characteristic equation

The equation \((r - 1)^2 = 0\) has one real root: \(r_1 = r_2 = 1\).
04

Write the general solution

Since we have one real root, the general solution is: \(y(x) = C_1 e^{x} + C_2 xe^{x}\). (b) Obtain the general solution for the differential equation: \(y'' + y' + 5y = 0\)
05

Identify coefficients

For this equation, we have the following coefficients: \(a = 1\), \(b = 1\), and \(c = 5\).
06

Form the characteristic equation

Our characteristic equation will be: \(r^2 + r + 5 = 0\).
07

Solve the characteristic equation

We will have two complex roots: \(r_{1,2} = \frac{-1 \pm \sqrt{-19}}{2}\).
08

Write the general solution

Since we have complex roots, our general solution is: \(y(t) = e^{-\frac{1}{2}t}\left(C_1 \cos{\frac{\sqrt{19}}{2}t} + C_2 \sin{\frac{\sqrt{19}}{2}t}\right)\). (c) Obtain the general solution for the differential equation: \(y'' + y' - 2y = 0\)
09

Identify coefficients

For this equation, we have the following coefficients: \(a = 1\), \(b = 1\), and \(c = -2\).
10

Form the characteristic equation

Our characteristic equation will be: \(r^2 + r - 2 = 0\).
11

Solve the characteristic equation

We will have two distinct real roots: \(r_1 = 1\) and \(r_2 = -2\).
12

Write the general solution

Since we have distinct real roots, our general solution is: \(y(x) = C_1 e^x + C_2 e^{-2x}\). (d) Obtain the general solution for the differential equation: \(y'' + 9y = 0\)
13

Identify coefficients

For this equation, we have the following coefficients: \(a = 1\), \(b = 0\), and \(c = 9\).
14

Form the characteristic equation

Our characteristic equation will be: \(r^2 + 9 = 0\).
15

Solve the characteristic equation

We will have two complex roots: \(r_1 = 3i\) and \(r_2 = -3i\).
16

Write the general solution

Our general solution is: \(y(x) = C_1 \cos{3x} + C_2 \sin{3x}\). (e) Obtain the general solution for the differential equation: \(y'' - 2y' = 0\)
17

Identify coefficients

For this equation, we have the following coefficients: \(a = 1\), \(b = -2\), and \(c = 0\).
18

Form the characteristic equation

Our characteristic equation will be: \(r^2 - 2r = 0\).
19

Solve the characteristic equation

We will have two distinct real roots: \(r_1 = 0\) and \(r_2 = 2\).
20

Write the general solution

Our general solution is: \(y(x) = C_1 + C_2 e^{2x}\). (f) Obtain the general solution for the differential equation: \(x'' - 16x = 0\)
21

Identify coefficients

For this equation, we have the following coefficients: \(a = 1\), \(b = 0\), and \(c = -16\).
22

Form the characteristic equation

Our characteristic equation will be: \(r^2 - 16 = 0\).
23

Solve the characteristic equation

We will have two distinct real roots: \(r_1 = 4\) and \(r_2 = -4\).
24

Write the general solution

Our general solution is: \(x(t) = C_1 e^{4t} + C_2 e^{-4t}\).

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Most popular questions from this chapter

Find a particular integral for the equation $$ \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} x}+y=1+x $$

If \(y=A \mathrm{e}^{2 x}\) is the general solution of \(\frac{\mathrm{d} y}{\mathrm{~d} x}=2 y\), find the particular solution satisfying \(y(0)=3\). What is the particular solution satisfying \(\frac{\mathrm{d} y}{\mathrm{~d} x}=2\) when \(x=0 ?\)

Use Euler's method to find a numerical solution of \(\frac{\mathrm{dy}}{\mathrm{d} x}=\frac{x y}{x^{2}+2}\) subject to \(y(1)=3\). Take \(h=0.1\) and hence approximate \(y(1.5)\). Obtain the true solution using the method of separation of variables. Work throughout to six decimal places.

The equation governing the flow of current \(i\) in a series \(L R\) circuit with applied constant voltage \(E\) is $$ L \frac{\mathrm{d} i}{\mathrm{~d} t}+R i=E $$ (a) Solve this equation subject to the condition \(i(0)=0 .\) (b) State the final value of the current. (c) Find the time taken for the current to reach \(95 \%\) of its final value.

Give one example each of a first-order linear equation, first-order non-linear equation, second-order linear equation, second-order non-linear equation.
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