Chapter 4: Q38E (page 200)

Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.
$t^{2} y'' + 3 ty' + y = t^{- 1}$

Short Answer

Expert verified

The solution of the given equation $t^{2} y'' + 3 ty' + y = t^{- 1}$ is localid="1664191735862" $y = - \frac{t^{- 1} lnt}{2} + \frac{1}{4 t^{3}} + c_{1} t^{- 1} + c_{2} t^{- 1} lnt$ .

Step by step solution

Substitute the values

Given differential equation is $t^{2} y'' + 3 ty' + y = t^{- 1}$

Letand 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:

$\begin{array}{rcl} t^{2} (r (r - 1) t^{r - 2} + 3 t ({rt}^{r - 1})) + t^{r} & = & 0 \\ (r^{2} + 2 r + 1) t^{r} & = & 0 \\ r^{2} + 2 r + 1 & = & 0 \\ {(r + 1)}^{2} & = & 0 \\ r & = & - 1 \end{array}$

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

Finding v1

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

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

And

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

Finding v2

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

Therefore,

$\begin{array}{rcl} y_{p} & = & \frac{2 lnt + 1}{4 t^{2}} (t^{- 1}) - \frac{1}{t^{2}} (t^{- 1} lnt) \\ = & - \frac{t^{- 1} lnt}{2} + \frac{1}{4 t^{3}} \end{array}$

Therefore, the total solution is $y = - \frac{t^{- 1} lnt}{2} + \frac{1}{4 t^{3}} + c_{1} t^{- 1} + c_{2} t^{- 1} lnt$ .

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Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.
$t^{2} y'' + 3 ty' + y = t^{- 1}$

Short Answer

Step by step solution

Substitute the values

Finding v1

Finding v2

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Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.t2y''+3ty'+y=t-1

Short Answer

Step by step solution

Substitute the values

Finding v1

Finding v2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

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Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find general solutions to the nonhomogeneous Cauchy-Euler equations using a variety of parameters.
$t^{2} y'' + 3 ty' + y = t^{- 1}$