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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: Q21E (page 199)

Devise a modification of the method for Cauchy-Euler equations to find a general solution to the given equation. ${(t - 2)}^{2} y'' (t) - 7 (t - 2) y' (t) + 7 y (t) = 0, 鈥� t > 2$

Short Answer

Expert verified

The general solution of the given equation ${(t - 2)}^{2} y'' (t) - 7 (t - 2) y' (t) + 7 y (t) = 0, 鈥� t > 2$ is $y = c_{1} (t - 2) + c_{2} {(t - 2)}^{7}$ .

Step by step solution

01

Substitute the values.

Given differential equation is ${(t - 2)}^{2} y'' (t) - 7 (t - 2) y' (t) + 7 y (t) = 0$

Let $t - 2 = u 鈬� dt = du$

Therefore, the equation becomes:

$u^{2} y'' (u) - 7 uy' (u) + 7 y (u) = 0$

Assume $y = u^{r}$ , then

$\begin{array}{l} y' = {ru}^{r - 1} \\ y'' = r (r - 1) u^{r - 2} \end{array}$

Substitute these equations in the differential equation;

$\begin{matrix} u^{2} r (r - 1) u^{r - 2} - 7 {uru}^{r - 1} + 7 u^{r} = 0 \\ (r (r - 1) - 7 r + 7) u^{r} = 0 \\ r^{2} - 8 r + 7 = 0 \end{matrix}$

02

Finding the roots of the auxiliary equation.

Find the roots of this equation.

$\begin{array}{l} r = \frac{8 卤 \sqrt{8^{2} - 4 脳 7 脳 1}}{2 脳 1} \\ r = \frac{8 卤 \sqrt{64 - 28}}{2} \\ r = \frac{8 卤 \sqrt{36}}{2} \\ r = \frac{8 卤 6}{2} \\ r = 1, 7 \end{array}$

Therefore, the general solution is $y = c_{1} u^{1} + c_{2} u^{7}$

Substitute $u = t - 2$ in the above solution, we get:

$y = c_{1} (t - 2) + c_{2} {(t - 2)}^{7}$

Thus, the solution is $y = c_{1} (t - 2) + c_{2} {(t - 2)}^{7} .$

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Most popular questions from this chapter

In Problems 33, use the method of undetermined coefficients to find a particular solution to the given higher-order equation. $y''' - y'' + y = sint$

Given that $y_{1} (t) = cost$ is a solution to $y'' - y' + y = sint$ and $y_{2} (t) = \frac{e^{2 t}}{3}$ is a solution to role="math" localid="1654926813168" $y'' - y' + y = e^{2 t}$ . Use the superposition principle to find solutions to the following differential equations:

$(a) 鈥夆赌夆赌夆赌� y'' - y' + y = 5 sint$

$(b) 鈥夆赌夆赌夆赌� y'' - y' + y = sint - 3 e^{2 t}$

$(c) 鈥夆赌夆赌夆赌� y'' - y' + y = 4 sint + 18 e^{2 t}$

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation.

$y'' + e^{t} y' + y = 7 + 3 t$

Find a particular solution to the given higher-order equation. $y^{4} - 3 y''' + 3 y'' - y' = 6 t - 20$

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

See all solutions

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