Chapter 4: Q41E (page 200)

Find a second linearly independent solution using reduction of order.
$t^{2} y'' - 2 ty' - 4 y = 0, t > 0; f (t) = t^{- 1}$ .

Short Answer

Expert verified

The second linearly independent solution of the given equation $t^{2} y'' - 2 ty' - 4 y = 0, t > 0; f (t) = t^{- 1}$ is $y = \frac{1}{5 t^{2}} - \frac{1}{20} + c_{1} t^{- 1} + c_{2} t^{4} lnt$ .

Step by step solution

Finding a homogeneous solution

Given differential equation is $t^{2} y'' - 2 ty' - 4 y = t^{- 1}$

Let $y = t^{r}$ and then find the solution to the associated homogeneous function;

$y' (t) = r t^{r - 1} y'' (t) = r (r - 1) t^{r - 2}$

Substitute these in the differential equation:

$t^{2} (r (r - 1) t^{r - 2}) - 2 t ({rt}^{r - 1}) - 4 t^{r} = 0 (r^{2} - 3 r - 4) t^{r} = 0 r^{2} - 3 r - 4 = 0 (r - 4) (r + 1) = 0$

$r = - 1$ and $r = 4$

So, the homogenous solution is $y = c_{1} t^{- 1} + c_{2} t^{4}$

Finding v1,v2

Now find the non-homogenous solution by using the variation of parameter method:

$\begin{array}{rcl} aW [y_{1}, y_{2}] & = & t^{2} ((t^{- 1} (4 t^{3}) - (- t^{- 2} (t^{4})) \\ = & t^{2} (5 t^{2}) \\ = & 5 t^{4} \end{array}$

And

$\begin{array}{rcl} v_{1} & = & \int - \frac{f (t) y_{2} (t)}{aW [y_{1}, y_{2}]} dt \\ = & \int - \frac{t^{- 2} t^{4}}{5 t^{4}} dt \\ = & \frac{1}{5 t} \end{array}$

And

$\begin{array}{rcl} v_{2} & = & \int \frac{f (t) y_{1} (t)}{aW [y_{1}, y_{2}]} dt \\ = & \int \frac{t^{- 2} t^{- 1}}{5 t^{2}} dt \\ = & - \frac{1}{20 t^{4}} \end{array}$

Hence,

$\begin{array}{rcl} y_{p} & = & \frac{1}{5 t} (t^{- 1}) - \frac{1}{20 t^{4}} (t^{4}) \\ = & \frac{1}{5 t^{2}} - \frac{1}{20} \end{array}$

Therefore, the total solution is $y = \frac{1}{5 t^{2}} - \frac{1}{20} + c_{1} t^{- 1} + c_{2} t^{4} lnt$

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Find a second linearly independent solution using reduction of order.
$t^{2} y'' - 2 ty' - 4 y = 0, t > 0; f (t) = t^{- 1}$ .

Short Answer

Step by step solution

Finding a homogeneous solution

Finding v1,v2

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Find a second linearly independent solution using reduction of order.t2y''-2ty'-4y=0,t>0;f(t)=t-1.

Short Answer

Step by step solution

Finding a homogeneous solution

Finding v1,v2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Decision Maths

Pure Maths

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find a second linearly independent solution using reduction of order.
$t^{2} y'' - 2 ty' - 4 y = 0, t > 0; f (t) = t^{- 1}$ .