Answered: find the general solution given that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

find the general solution given that y1(t)=t is a solution. use the method of reduction of order

t^2y''+2ty'-2y=0

Expert Solution

Step 1

To find a solution to the differential equation $t^{2} y'' + 2 t y' - 2 y = 0 \dots (1)$ given that $y_{1} (t) = t$ is a solution.

Assume $y_{2} (t) = t y_{1} (t)$ is a solution to 1, for a proper choice of $v (t)$ .

Here,

$\begin{array}{rcl} y_{2}' (t) & = & v (t) + v' (t) t \\ y_{2}'' (t) & = & v' (t) + v' (t) + v'' (t) t \end{array}$

Substituting these in 1, we get,

$\begin{array}{rcl} t^{2} (2 v' (t) + v'' (t) t) + 2 t (v (t) + v' (t) t) - 2 (v (t) t) & = & 0 \\ 2 t^{2} v' (t) + t^{3} v'' (t) + 2 t v (t) + 2 t^{2} v' (t) - 2 t v (t) & = & 0 \\ 4 t^{2} v' (t) + t^{3} v'' (t) & = & 0 \end{array}$

Take $v' (t) = w (t)$ and $v'' (t) = w' (t)$ .

Then we get,

$4 t^{2} w (t) + t^{3} w' (t) = 0$

Dividing by $t^{3}$ , results in,

$w' (t) + \frac{4}{t} w' (t) = 0$ $\dots (2)$

which is a linear first order homogeneous differential equation.

Now solve 2 to find $w (t)$ :

First find the integrating factor:

The integrating factor is given by:

$\begin{array}{rcl} I F & = & e^{\int \frac{4}{t} d t} \\ = & e^{4 \ln t} \end{array}$

Therefore,

$\frac{d}{d t} (e^{4 \ln t} w (t)) = 0$

Integrating on both sides, we get,

$\begin{array}{rcl} e^{4 \ln t} w (t) & = & c \\ w (t) & = & \frac{c}{e^{4 \ln t}} \\ w (t) & = & c e^{- 4 \ln t} \end{array}$

Now $w (t) = v' (t)$

Therefore,

$\begin{array}{rcl} v (t) & = & \int w (t) d t \\ = & \int c e^{- 4 \ln t} d t \\ = & c \frac{e^{- 4 \ln t}}{\frac{- 4}{t}} + k \\ = & \frac{- 1}{4} c t e^{- 4 \ln t} + k \end{array}$

Take $c = - 4$ and $k = 0$ .

Then,

$v (t) = t e^{- 4 \ln t}$

Therefore,

$y_{2} (t) = t^{2} e^{- 4 \ln t}$

And the general solution of 1 is,

$y (t) = c_{1} t + c_{2} t^{2} e^{- 4 \ln t}$

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ISBN:

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Wiley, John & Sons, Incorporated