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Fundamentals Of Differential Equations And Boundary Value Problems

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

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 4: Q6E (page 186)

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

Short Answer

Expert verified

The general solution of the given differential equation is $y = c_{1} e^{- 2 x} + c_{2} e^{- 3 x} + e^{x} + x^{2}$ .

Step by step solution

01

Write the auxiliary equation of the given differential equation.

The differential equation is,

$y'' + 5 y' + 6 y = 6 x^{2} + 10 x + 2 + 12 e^{x} 鈥夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌� 鈥� (1)$

Write the homogeneous differential equation of the equation (1),

$y'' + 5 y' + 6 y = 0$

The auxiliary equation for the above equation,

$m^{2} + 5 m + 6 = 0$

02

Now find the complementary solution of the given equation is

Solve the auxiliary equation,

$\begin{matrix} m^{2} + 5 m + 6 = 0 \\ m^{2} + 3 m + 2 m + 6 = 0 \\ m (m + 3) + 2 (m + 3) = 0 \\ (m + 2) (m + 3) = 0 \end{matrix}$

The roots of the auxiliary equation are,

$m_{1} = - 2, 鈥夆赌夆赌� & 鈥夆赌夆赌� m_{2} = - 3$

The complementary solution of the given equation is,

$y_{c} = c_{1} e^{- 2 x} + c_{2} e^{- 3 x}$

03

Use the given particular solution to find a general solution for the equation.

The given particular solution,

$y_{p} (x) = e^{x} + x^{2}$

Therefore, the general solution is,

$\begin{matrix} y = y_{c} (x) + y_{p} (x) \\ y = c_{1} e^{- 2 x} + c_{2} e^{- 3 x} + e^{x} + x^{2} \end{matrix}$

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Most popular questions from this chapter

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

Find a particular solution to the differential equation. $4 y'' + 11 y' - 3 y = - 2 {te}^{- 3 t}$

Determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) $y'' - y' - 12 y = 2 t^{6} e^{- 3 t}$

Find a particular solution to the differential equation.

$y'' + 2 y' - y = 10$

Solve the given initial value problem. $w'' - 4 w' + 2 w = 0;$ $w (0) = 0,$ $w' (0) = 1$

See all solutions

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