Chapter 8: Q15P (page 443)

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.
$y^{'}' + 9 y = \cos 3 t, y_{0} = 0, y_{0}^{'} = 6$

Short Answer

Expert verified

The given differential equation's solution is $y = \frac{1}{6} t \sin 2 t + 2 \sin 3 t$ .

Step by step solution

Given information from question

The differential equation is $y^{'}' + 9 y = \cos 3 t, y_{0} = 0, y_{0}^{'} = 6 .$ .

Laplace transform

Application of Laplace transform:

It's used to simplify complicated differential equations by using polynomials. It's used to convert derivatives into numerous domain variables, then utilise the Inverse Laplace transform to convert the polynomials back to the differential equation.

Take the Laplace of given equation

The given equation is

$y " + 9 y = c o s 3 t$

Take the Laplace of equation

$L \{y^{'}' + 9 y\} = L \{\cos 3 t\} L \{y "\} + 9 L \{y\} = L \{\sin t\} p^{2} L \{y\} - {py}_{o} - y'_{0} + 9 L \{y\} = \frac{p}{p^{2} + 9} (p^{2} + 9) L \{y\} - {py}_{o} - y'_{0} = \frac{p}{p^{2} + 9}$

Further solve

$(p^{2} + 9) L \{y\} - p (0) - 6 = \frac{p}{p^{2} + 9} (p^{2} + 9) L \{y\} = \frac{p}{p^{2} + 9} + 6 L \{y\} = \frac{p}{(p^{2} + 9) {(p^{2} + 9)}^{2}} + \frac{6}{(p^{2} + 9)} L \{y\} = \frac{p}{{(p^{2} + 3^{2})}^{2}} + \frac{6}{(p^{2} + 3^{2})}$

The inverse Laplace is

$y = L^{- 1} \{\frac{p}{{(p^{2} + 3^{2})}^{2}}\} + L^{- 1} \{\frac{6}{(p^{2} + 3^{2})}\} y = \frac{1}{6} t \sin 2 t + \frac{6}{3} \sin 3 t y = \frac{1}{6} t \sin 2 t + 2 \sin 3 t$

Thus, the given differential equation's solution is $y = \frac{1}{6} t \sin 2 t + 2 \sin 3 t$ .

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By using Laplace transforms, solve the following differential equations subject to the given initial conditions.
$y^{'}' + 9 y = \cos 3 t, y_{0} = 0, y_{0}^{'} = 6$

Short Answer

Step by step solution

Given information from question

Laplace transform

Take the Laplace of given equation

One App. One Place for Learning.

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By using Laplace transforms, solve the following differential equations subject to the given initial conditions.y''+9y=cos3t, y0=0,y0'=6

Short Answer

Step by step solution

Given information from question

Laplace transform

Take the Laplace of given equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Mechanics and Materials

Circular Motion and Gravitation

Magnetism and Electromagnetic Induction

Torque and Rotational Motion

Kinematics Physics

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions.
$y^{'}' + 9 y = \cos 3 t, y_{0} = 0, y_{0}^{'} = 6$