Q14E Find a general solution to the g... [FREE SOLUTION]

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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 4: Q14E (page 199)

Find a general solution to the given Cauchy-Euler equation for $t > 0$
$t^{2} y'' (t) - 3 ty' (t) + 4 y (t) = 0$

Short Answer

Expert verified

The general solution to the given equation $t^{2} y'' (t) - 3 ty' (t) + 4 y (t) = 0$ is $y (t) = c_{1} t^{2} + c_{2} t^{2} lnt$ .

Step by step solution

Check the associated characteristic equation.

For a differential equation ${at}^{2} y'' (t) + bty' (t) + cy (t) = f (t)$ , we have that the associated characteristic equation is ${ar}^{2} + (b - a) r + c = 0$ .

In our case, the coefficient of $t^{2} y'' (t)$ is $a = 1$ , the one multiplying is $ty' (t)$ is $b = - 3$ and the one multiplying $y (t)$ is $c = 4$ , therefore the associated characteristic equation is:

$\begin{array}{l} r^{2} + (- 3 - 1) r + 4 = 0 \\ r^{2} - 4 r + 4 = 0 \\ {(r - 2)}^{2} = 0 \\ r = 2, 2 \end{array}$

Check the differential equation.

When one has a double root to the associated characteristic equation, linearly independent solutions to the given differential equation are $y_{1} (t) = t^{r}$ and $y_{2} (t) = t^{r} lnt$ .

So, we have $y_{1} (t) = t^{2}$ , $y_{2} (t) = t^{2} lnt$ and hence the general solution to the given equation is $y (t) = c_{1} t^{2} + c_{2} t^{2} lnt$ .

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91影视

Find a general solution to the given Cauchy-Euler equation for $t > 0$
$t^{2} y'' (t) - 3 ty' (t) + 4 y (t) = 0$

Short Answer

Step by step solution

Check the associated characteristic equation.

Check the differential equation.

One App. One Place for Learning.

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91影视

Find a general solution to the given Cauchy-Euler equation for t>0t2y''(t)-3ty'(t)+4y(t)=0

Short Answer

Step by step solution

Check the associated characteristic equation.

Check the differential equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Mechanics Maths

Probability and Statistics

Calculus

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find a general solution to the given Cauchy-Euler equation for $t > 0$
$t^{2} y'' (t) - 3 ty' (t) + 4 y (t) = 0$