Chapter 4: Q18E (page 199)

Find a general solution for $t < 0$ . $t^{2} y'' (t) + 3 ty' (t) + 5 y (t) = 0$

Short Answer

Expert verified

The general solution of the given equation $t^{2} y'' (t) + 3 ty' (t) + 5 y (t) = 0$ is $y = c_{1} t^{- 1} \cos (2 lnt) + c_{2} t^{- 1} \sin (2 lnt)$ for $t < 0$ .

Step by step solution

Substitute the values.

Given differential equation is $t^{2} y'' (t) + 3 ty' (t) + 5 y (t) = 0$

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

$\begin{array}{l} y' = {rt}^{r - 1} \\ y'' = r (r - 1) t^{r - 2} \end{array}$

Substitute these equations in the differential equation;

$\begin{matrix} t^{2} r (r - 1) t^{r - 2} + 3 {trt}^{r - 1} + 5 t^{r} = 0 \\ (r (r - 1) + 3 r + 5) t^{t} = 0 \\ r^{2} + 2 r + 5 = 0 \end{matrix}$

The auxiliary equation is:

$r^{2} + 2 r + 5 = 0$

Finding the roots of the auxiliary equation.

Find the roots of the auxiliary equation.

$\begin{matrix} r = \frac{- 2 \pm \sqrt{2^{2} - 4 \times 5 \times 1}}{2 \times 1} \\ r = \frac{- 2 \pm \sqrt{4 - 20}}{2} \\ r = \frac{- 2 \pm \sqrt{- 16}}{2} \\ r = - 1 \pm 2 i \end{matrix}$

Write the general solution.

When the roots to the characteristic equation are complex, $r = α \pm βi$ and if $t > 0$ then the linearly independent solutions are $y_{1} (t) = t^{α} \cos (βlnt)$ and $y_{2} (t) = t^{α} \sin (βlnt) .$ But if we have that $t < 0$ , then the linearly independent solutions are given as:

$y_{1} (t) = (- t)^{α} \cos (βln (- t) and y_{2} (t) = (- t)^{α} \sin (βln (- t))$

Therefore, the general solution is $y = c_{1} t^{- 1} \cos (2 \ln (- t)) + c_{2} t^{- 1} \sin (2 \ln (- t)) .$

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Find a general solution for $t < 0$ . $t^{2} y'' (t) + 3 ty' (t) + 5 y (t) = 0$

Short Answer

Step by step solution

Substitute the values.

Finding the roots of the auxiliary equation.

Write the general solution.

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Find a general solution for t<0.t2y''(t)+3ty'(t)+5y(t)=0

Short Answer

Step by step solution

Substitute the values.

Finding the roots of the auxiliary equation.

Write the general solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

Probability and Statistics

Decision Maths

Applied Mathematics

Calculus

Study anywhere. Anytime. Across all devices.

Find a general solution for $t < 0$ . $t^{2} y'' (t) + 3 ty' (t) + 5 y (t) = 0$