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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset Vaia Explanations$ Math

9th Edition · 616 pages

ISBN: 9780321977069

Chapter 4: Q2RP (page 231)

Find a general solution to the given differential equation. $49 y'' + 14 y' + y = 0$

Short Answer

Expert verified

$y = c_{1} e^{\frac{- 1}{7} t} + c_{2} {te}^{\frac{- 1}{7} t}$

Step by step solution

01

Write the auxiliary equation of the given differential equation

The given differential equation is $49 y'' + 14 y' + y = 0 .$

The auxiliary equation for the above equation $49 m^{2} + 14 m + 1 = 0 .$

02

Find the roots of the auxiliary equation

Solve the auxiliary equation,

$\begin{matrix} 49 m^{2} + 14 m + 1 = 0 \\ 49 m^{2} + 7 m + 7 m + 1 = 0 \\ 7 m (7 m + 1) + 1 (7 m + 1) = 0 \\ (7 m + 1) (7 m + 1) = 0 \end{matrix}$

The roots of the auxiliary equation are $m_{1} = \frac{- 1}{7} & m_{2} = \frac{- 1}{7} .$

Thus, the general solution of the given equation is $y = c_{1} e^{\frac{- 1}{7} t} + c_{2} {te}^{\frac{- 1}{7} t} .$

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Most popular questions from this chapter

Swinging Door. The motion of a swinging door with an adjustment screw that controls the amount of friction on the hinges is governed by the initial value problem

$Iθ'' + bθ' + kθ = 0; θ (0) = θ_{0}, θ' (0) = v_{0}$ ,

where $θ$ is the angle that the door is open, $I$ is the moment of inertia of the door about its hinges, $b > 0$ is a damping constant that varies with the amount of friction on the door, $k > 0$ is the spring constant associated with the swinging door, $θ_{0}$ is the initial angle that the door is opened, and $v_{0}$ is the initial angular velocity imparted to the door (see figure). If $I$ and $k$ are fixed, determine for which values of $b$ the door will not continually swing back and forth when closing.

The auxiliary equation for the given differential equation has complex roots. Find a general solution $4 y'' - 4 y' + 26 y = 0$ .

Find a general solution to the differential equation. $y'' (x) + 6 y' (x) + 10 y (x) = 10 x^{4} + 24 x^{3} + 2 x^{2} - 12 x + 18$

Find a general solution to the differential equation.

$y'' + 4 y = sinθ - cosθ$

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. $y'' + 5 y' + 6 y = 6 x^{2} + 10 x + 2 + 12 e^{x}, y_{p} (x) = e^{x} + x^{2}$

See all solutions

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