Chapter 4: Q19E (page 199)

Solve the given initial value problem for the Cauchy-Euler equation.
$t^{2} y'' (t) - 4 ty' (t) + 4 y (t) = 0; y (1) = - 2, y' (1) = - 11$

Short Answer

Expert verified

The solution of the given initial value problem $t^{2} y'' (t) - 4 ty' (t) + 4 y (t) = 0; y (1) = - 2, y' (1) = - 11$ is $y = - 3 t^{4} + t$ .

Step by step solution

Substitute the values.

Given differential equation is $t^{2} y'' (t) - 4 ty' (t) + 4 y (t) = 0$

Assume $y = t^{r}$ , then we have;

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

Substitute these equations in the differential equation;

$\begin{matrix} t^{2} r (r - 1) t^{r - 2} - 4 {trt}^{r - 1} + 4 t^{r} = 0 \\ (r (r - 1) - 4 r + 4) t^{t} = 0 \\ r^{2} - 5 r + 4 = 0 \end{matrix}$

The auxiliary equation is $r^{2} - 5 r + 4 = 0$ .

Finding the roots of the auxiliary equation.

Find the roots of this equation:

$\begin{matrix} r = \frac{5 \pm \sqrt{5^{2} - 4 \times 4 \times 1}}{2 \times 1} \\ r = \frac{5 \pm \sqrt{25 - 16}}{2} \\ r = \frac{5 \pm \sqrt{9}}{2} \\ r = \frac{5 \pm 3}{2} \\ r = 4, 1 \end{matrix}$

Hence, the general solution is $y = c_{1} t^{4} + c_{2} t^{1} .$

Finding the values of c1,c2

Using the given initial conditions;

$\begin{array}{l} y (1) = c_{1} {(1)}^{4} + c_{2} (1) \\ - 2 = c_{1} + c_{2} \\ c_{1} + c_{2} = - 2 \dots (1) \end{array}$

And we have $y' (t) = 4 c_{1} t^{3} + c_{2}$ then:

$\begin{array}{l} y' (1) = 4 c_{1} {(1)}^{3} + c_{2} \\ 4 c_{1} + c_{2} = - 11 \dots (2) \end{array}$

Subtract (2) from (1), we get:

$\begin{array}{l} 3 c_{1} = - 9 \\ c_{1} = - 3 \end{array}$

Therefore,

$c_{2} = 1$

Thus, the solution is $y = - 3 t^{4} + t .$

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Solve the given initial value problem for the Cauchy-Euler equation.
$t^{2} y'' (t) - 4 ty' (t) + 4 y (t) = 0; y (1) = - 2, y' (1) = - 11$

Short Answer

Step by step solution

Substitute the values.

Finding the roots of the auxiliary equation.

Finding the values of c1,c2

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Solve the given initial value problem for the Cauchy-Euler equation.t2y''(t)-4ty'(t)+4y(t)=0;y(1)=-2, y'(1)=-11

Short Answer

Step by step solution

Substitute the values.

Finding the roots of the auxiliary equation.

Finding the values of c1,c2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Statistics

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

Solve the given initial value problem for the Cauchy-Euler equation.
$t^{2} y'' (t) - 4 ty' (t) + 4 y (t) = 0; y (1) = - 2, y' (1) = - 11$