Chapter 4: Q21E (page 199)

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

Short Answer

Expert verified

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

Step by step solution

Substitute the values.

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

Let $t - 2 = u \Rightarrow dt = du$

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;

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

Finding the roots of the auxiliary equation.

Find the roots of this equation.

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

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

Short Answer

Step by step solution

Substitute the values.

Finding the roots of the auxiliary equation.

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

Short Answer

Step by step solution

Substitute the values.

Finding the roots of the auxiliary equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

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Calculus

Study anywhere. Anytime. Across all devices.

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