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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 6: Q2E (page 332)

Find a general solution for the differential equation with x as the independent variable.
$y''' - 3 y'' - y' + 3 y = 0$

Short Answer

Expert verified

Thus, the general solution to the given differential equation is; $y = C_{1} e^{3 x} + C_{2} e^{x} + C_{3} e^{- x}$

Step by step solution

01

Find the auxiliary equation

The given differential equation is;

$y''' - 3 y'' - y' + 3 y = 0$

The auxiliary equation is,

$m^{3} - 3 m^{2} - m + 3 = 0$

Take a fraction of the above equation,

$\begin{matrix} m^{2} (m - 3) - 1 (m - 3) = 0 \\ (m - 3) (m^{2} - 1) = 0 \\ m_{1} = 3, m_{2} = 1, m_{3} = - 1 \end{matrix}$

02

Write the general solution

The roots are real and distinct; therefore the general solution to the given differential equation is given as:

$\begin{matrix} y = C_{1} e^{m_{1} x} + C_{2} e^{m_{2} x} + C_{3} e^{m_{3} x} \\ y = C_{1} e^{(3) x} + C_{2} e^{(1) x} + C_{3} e^{(- 1) x} \\ y = C_{1} e^{3 x} + C_{2} e^{x} + C_{3} e^{- x} \end{matrix}$

Thus, the general solution to the given differential equation is;

$y = C_{1} e^{3 x} + C_{2} e^{x} + C_{3} e^{- x}$

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Most popular questions from this chapter

On a smooth horizontal surface, a mass of m₁ kg isattached to a fixed wall by a spring with spring constantk₁ N/m. Another mass of m₂ kg is attached to thefirst object by a spring with spring constant k₂ N/m. Theobjects are aligned horizontally so that the springs aretheir natural lengths. As we showed in Section 5.6, thiscoupled mass–spring system is governed by the systemof differential equations

$m_{1} \frac{d^{2} x}{d t^{2}} + (k_{1} + k_{2}) x - k_{2} y = 0$

$m_{2} \frac{d^{2} y}{d t^{2}} - k_{2} x + k_{2} y = 0$

Let’s assume that m₁ = m₂ = 1, k₁ = 3, and k₂ = 2.If both objects are displaced 1 m to the right of theirequilibrium positions (compare Figure 5.26, page 283)and then released, determine the equations of motion forthe objects as follows:

(a)Show that x(2) satisfies the equation $x^{(4)} (t) + 7 x'' (t) + 6 x (t) = 0$

(b) Find a general solution x(2) to (36).

(c) Substitute x(2) back into (34) to obtain a generalsolution for y(2)

(d) Use the initial conditions to determine the solutions,x(2) and y(2), which are the equations of motion.

Find a general solution for the differential equation with x as the independent variable:

$2 y''' + 5 y'' - 13 y' + 7 y = 0$

Deflection of a Beam Under Axial Force. A uniform beam under a load and subject to a constant axial force is governed by the differential equation

$y^{(4)} (x) - k^{2} y^{''} (x) = q (x), 0 < x < L,$

where is the deflection of the beam, L is the length of the beam, k²is proportional to the axial force, and q(x) is proportional to the load (see Figure 6.2).

(a) Show that a general solution can be written in the form

$y (x) = C_{1} + C_{2} x + C_{3} e^{k x} + C_{4} e^{- k x} + \frac{1}{k^{2}} \int q (x) x d x - \frac{x}{k^{2}} \int q (x) d x + \frac{e^{k x}}{2 k^{3}} \int q (x) e^{- k x} d x - \frac{e^{- k x}}{2 k^{3}} \int q (x) e^{k x} d x$

(b) Show that the general solution in part (a) can be rewritten in the form

$y (x) = c_{1} + c_{2} x + c_{3} e^{k x} + c_{4} e^{- k x} + \int_{0}^{x} q (s) G (s, x) d s,$

where

$G (s, x) : = \frac{s - x}{k^{2}} - \frac{s i n h [k (s - x)]}{k^{3}} .$

(c) Let q(x)=1 First compute the general solution using the formula in part (a) and then using the formula in part (b). Compare these two general solutions with the general solution

$y (x) = B_{1} + B_{2} x + B_{3} e^{k x} + B_{4} e^{- k x} - \frac{1}{2 k^{2}} x^{2},$

which one would obtain using the method of undetermined coefficients.

find a general solution to the given equation.

$y''' + 4 y'' + y' - 26 y = e^{- 3 x} s i n 2 x + x$

Find a general solution to the givenhomogeneous equation. $(D + 4) (D - 3) {(D + 2)}^{3} {(D^{2} + 4 D + 5)}^{2} D^{5} [y] = 0$

See all solutions

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