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Chapter 8: Problem 3

Identify each of the differential equations as type (for example, separable, linear first order, linear second order, etc.), and then solve it. $$y^{\prime \prime \prime}+2 y^{\prime \prime}+2 y^{\prime}=0$$

Short Answer

Expert verified
The given differential equation is a third-order homogeneous linear differential equation. The general solution is \[ y(t) = C_1 + e^{-t }( C_2 cos(t) + C_3 sin(t) )\]

Step by step solution

01

Identify the type of the differential equation

The given differential equation is a third-order homogeneous linear differential equation: \[ y^{\text{'''} } + 2 y^{\text{''} } + 2 y^{\text{'}} = 0 \]
02

Form the characteristic equation

For a homogeneous linear differential equation of the form \[ a_n y^{(n)} + a_{n-1} y^{(n-1)} + \tdots + a_1 y' + a_0 y = 0 \], the characteristic equation is obtained by replacing each derivative term with a power of \(r\). Thus, the characteristic equation for \[ y^{\text{'''}} + 2 y^{\text{''}} + 2 y^{\text{'}} = 0 \] is: \[ r^3 + 2r^2 + 2r = 0 \]
03

Factor the characteristic equation

To find the roots, first factor out the common factor:\[ r (r^2 + 2r + 2) = 0 \]
04

Solve the factored equation

Set each factor equal to zero to find the roots:\[ r = 0 \] and solve the quadratic equation: \[ r^2 + 2r + 2 = 0 \] Using the quadratic formula \[ r = \frac{-b 卤 鈭歿b^2 - 4ac}}{2a} \] where \( a = 1, b = 2, c = 2 \):\[ r = \frac{-2 卤鈭歿2^2 - 4 (1)(2)}}{2 (1)} = \frac{-2 卤鈭歿4 - 8}}{2} = \frac{-2 卤 鈭歿-4}}{2} = \frac{-2 卤 2i}{2} = -1 卤 i \] So the roots are: \[ r = 0, -1 + i, -1 - i \]
05

Write the general solution

Given that the roots are real and complex, the general solution form for the differential equation with roots \(r_1, \alpha 卤 i \beta\): \[ y = C_1 e^{r_1 t} + e^{ 伪 t} ( C_2 cos( 尾 t ) + C_3 sin( 尾 t ) ) \] For the roots \( 0, -1+i\), and \( -1-i \): \[ y(t) = C_1 + e^{-t} ( C_2 cos( t ) + C_3 sin( t ) ) \]}

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Homogeneous Linear Differential Equation
A homogeneous linear differential equation is a type of differential equation where the dependent variable and all its derivatives appear linearly. This means you won't see any powers or products of the dependent variable or its derivatives in the equation.

The general form is:

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Most popular questions from this chapter

A mechanical or electrical system is described by the differential equation \(y^{\prime \prime}+\omega^{2} y=f(t) .\) Find \(y\) if \(f(t)=\left\\{\begin{array}{ll}1, & 0a\) separately, remembering that \(f(t)=0\) for \(t>a .\) Show that \(y=\left\\{\begin{array}{ll}\frac{1}{\omega^{2}}(1-\cos \omega t), & ta.\end{array}\right.\) Sketch the motion if \(a=\frac{1}{3} T\) where \(T\) is the period for free vibrations of the system; if \(a=\frac{3}{2} T ;\) if \(a=\frac{1}{10} T\).

The differential equation for the path of a planet around the sun (or any object in an inverse square force field) is, in polar coordinates, \(\frac{1}{r^{2}} \frac{d}{d \theta}\left(\frac{1}{r^{2}} \frac{d r}{d \theta}\right)-\frac{1}{r^{3}}=-\frac{k}{r^{2}}\). Make the substitution \(u=1 / r\) and solve the equation to show that the path is a conic section.

For each of the following differential equations, separate variables and find a solution containing one arbitrary constant. Then find the value of the constant to give a particular solution satisfying the given boundary condition. Computer plot a slope field and some of the solution curves. \(y^{\prime}+2 x y^{2}=0, \quad y=1\) when \(x=2\)

Evaluate each of the following definite integrals by using the Laplace transform table. $$\int_{0}^{\infty} t e^{-t} \sin 5 t d t$$

Use the Laplace transform table to find \(f(t)=\int_{0}^{t} e^{-\tau} \sin (t-\tau) d \tau .\) Hint: In \(L 34\) let \(g(t)=e^{-t}\) and \(h(t)=\sin t,\) and find \(G(p) H(p)\) which is the Laplace transform of the integral you want. Break the result into partial fractions and look up the inverse transforms.
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