Chapter 4: Q22E (page 199)

Devise a modification of the method for Cauchy-Euler equations to find a general solution to the given equation. ${(t + 1)}^{2} y'' (t) + 10 (t + 1) y' (t) + 14 y (t) = 0, t > - 1$

Short Answer

Expert verified

The general solution to the given equation ${(t + 1)}^{2} y'' (t) + 10 (t + 1) y' (t) + 14 y (t) = 0, t > - 1$ is $y = c_{1} {(t + 1)}^{- 2} + c_{2} {(t + 1)}^{- 7}$ .

Step by step solution

Substitute the values:

Given differential equation is ${(t + 1)}^{2} y'' (t) + 10 (t + 1) y' (t) + 14 y (t) = 0$

Let $t + 1 = u \Rightarrow dt = du$

Therefore, the equation becomes:

$u^{2} y'' (u) + 10 uy' (u) + 14 y (u) = 0$

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;

$\begin{matrix} u^{2} r (r - 1) u^{r - 2} + 10 {uru}^{r - 1} + 14 u^{r} = 0 \\ (r (r - 1) + 10 r + 14) u^{r} = 0 \\ r^{2} + 9 r + 14 = 0 \end{matrix}$

The auxiliary equation is:

$r^{2} + 9 r + 14 = 0$

Finding roots

Find the roots of this equation.

$\begin{matrix} r = \frac{- 9 \pm \sqrt{9^{2} - 4 \times 14 \times 1}}{2 \times 1} \\ r = \frac{- 9 \pm \sqrt{81 - 56}}{2} \\ r = \frac{- 9 \pm \sqrt{25}}{2} \\ r = - 2, - 7 \end{matrix}$

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

Substitute $u = t + 1$ in the above solution, we get $y = c_{1} {(t + 1)}^{- 2} + c_{2} {(t + 1)}^{- 7}$ .

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

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Devise a modification of the method for Cauchy-Euler equations to find a general solution to the given equation. ${(t + 1)}^{2} y'' (t) + 10 (t + 1) y' (t) + 14 y (t) = 0, t > - 1$

Short Answer

Step by step solution

Substitute the values:

Finding roots

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Devise a modification of the method for Cauchy-Euler equations to find a general solution to the given equation.(t+1)2y''(t)+10(t+1)y'(t)+14y(t)=0,t>-1

Short Answer

Step by step solution

Substitute the values:

Finding roots

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Probability and Statistics

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Statistics

Discrete Mathematics

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. ${(t + 1)}^{2} y'' (t) + 10 (t + 1) y' (t) + 14 y (t) = 0, t > - 1$