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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: Q4.3-11E (page 172)

Find a general solution $z " + 10 z' + 25 z = 0$

Short Answer

Expert verified

The general solution of the given equation $z " + 10 z' + 25 z = 0$ is $y (t) = (c_{1} + c_{2} t) e^{- 5 t}$ .

Step by step solution

01

Given information.

Given differential equation is $z " + 10 z' + 25 z = 0$ .

02

Finding roots of the auxiliary equation.

Then the auxiliary equation is $r^{2} + 10 r + 25 = 0$

Solve the auxiliary equation to obtain the roots.

${(r + 5)}^{2} = 0 r + 5 = 0 r = - 5$

03

Final answer.

Therefore, the general solution is $y (t) = (c_{1} + c_{2} t) e^{- 5 t}$

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Most popular questions from this chapter

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

Find a general solution to the differential equation.

y'' - 2 y' - 3 y = 3 t^{2} - 5

Find a general solution. $y'' - 8 y' + 7 y = 0$

$RLC$ Series Circuit. In the study of an electrical circuit consisting of a resistor, capacitor, inductor, and an electromotive force (see Figure), we are led to an initial value problem of the form

$\begin{array}{r} (20) L \frac{dI}{dt} + RI + \frac{q}{C} = E (t); \\ q (0) = q_{0} \\ I (0) = I_{0}, \end{array}$

where $L$ is the inductance in henrys, $R$ is the resistance in ohms, $C$ is the capacitance in farads, $E (t)$ is the electromotive force in volts, $q (t)$ is the charge in coulombs on the capacitor at the time $t$ , androle="math" localid="1654852406088" $I = dq / dt$ is the current in amperes. Find the current at time t if the charge on the capacitor is initially zero, the initial current is zero,role="math" localid="1654852401965" $L = 10 H, R = 20 Ω, C = (6260)^{- 1} F$ , androle="math" localid="1654852397693" $E (t) = 100 V$ .

Find a particular solution to the differential equation. $y'' + 3 y = - 9$

See all solutions

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