Chapter 4: Q24E (page 172)

Solve the given initial value problem. $y'' + 9 y = 0;$ $y (0) = 1,$ $y' (0) = 1$

Short Answer

Expert verified

The solution of the given initial value $y'' + 9 y = 0$ is $y (t) = \cos 3 t + \frac{}{} \sin 3 t$ when $y (0) = 1$ and $y' (0) = 1$ .

Step by step solution

Complex conjugate roots.

If the auxiliary equation has complex conjugate roots $α \pm iβ$ , then the general solution is given as:

$y (t) = c_{1} e^{αt} cosβt + c_{2} e^{αt} sinβt$

Finding the roots of the auxiliary equation.

Given differential equation is $y'' + 9 y = 0$

Then the auxiliary equation is;

$\begin{matrix} r^{2} + 9 = 0 \\ r^{2} = - 9 \\ r = \pm \sqrt{- 9} \\ r = \pm 3 i \end{matrix}$

Therefore, the general solution is:

$\begin{array}{l} y (t) = e^{0 \times t} (c_{1} \cos (3 t) + c_{2} \sin (3 t)) \\ = c_{1} \cos (3 t) + c_{2} \sin (3 t)) \end{array}$

Finding the values of c1 and c2

Given initial conditions are $y (0) = 1$ and $y' (0) = 1$

$\begin{array}{l} y (0) = (c_{1} \cos (3 \times 0) + c_{2} \sin (3 \times 0)) \\ c_{1} = 1 \end{array}$

And $y' (t) = (- c_{1} \sin (3 t) + c_{2} \cos (3 t))$

Then $y' (0) = (- 3 c_{1} \sin (3 \times 0) + 3 c_{2} \cos (3 \times 0))$

role="math" localid="1654843939522" $\begin{matrix} 3 c_{2} = 1 \\ c_{2} = \frac{1}{3} \end{matrix}$

Therefore, the solution is role="math" localid="1654843966518" $y (t) = \cos 3 t + \frac{1}{3} \sin 3 t$

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Solve the given initial value problem. $y'' + 9 y = 0;$ $y (0) = 1,$ $y' (0) = 1$

Short Answer

Step by step solution

Complex conjugate roots.

Finding the roots of the auxiliary equation.

Finding the values of c1 and c2

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Solve the given initial value problem.y''+9y=0;y(0)=1,y'(0)=1

Short Answer

Step by step solution

Complex conjugate roots.

Finding the roots of the auxiliary equation.

Finding the values of c1 and c2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Probability and Statistics

Statistics

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Solve the given initial value problem. $y'' + 9 y = 0;$ $y (0) = 1,$ $y' (0) = 1$