Chapter 4: Q4E (page 157)

Verify that $y = s i n 3 t + 2 c o s 3 t$ is a solution to the initial value problem
$2 y'' + 18 y = 0; y (0) = 2, y' (0) = 3$
Find the maximum of $| y (t) |$ for $- \infty < t < \infty$ .

Short Answer

Expert verified

Here $y = s i n 3 t + 2 c o s 3 t$ is a solution for the given differential equation, the maximum value $|y| = 2.236$ .

Step by step solution

Differentiate the value of y

To verify that $y = s i n 3 t + 2 c o s 3 t$ is a solution of $2 y'' + 18 y = 0$ .

Find $y'$ and $y'' .$

role="math" localid="1664098240818" $y' (t) = 3 c o s 3 t - 6 s i n 3 t y'' (t) = - 9 s i n 3 t - 18 c o s 3 t$

Substitute these two equations in the differential equation;

$2 (- 9 s i n 3 t - 18 c o s 3 t) + 18 (s i n 3 t + 2 c o s 3 t) = 0 - 18 s i n 3 t - 36 c o s 3 t + 18 s i n 3 t + 36 c o s 3 t = 0 0 = 0$

Therefore, LHS = RHS.

Verify boundary condition.

Now verify the boundary conditions;

$y (0) = s i n (3 \times 0) + 2 c o s (3 \times 0) = 2$

And

$y' (0) = 3 c o s (3 \times 0) - 6 s i n (3 \times 0) = 3$

Therefore, $y = s i n 3 t + 2 c o s 3 t$ is a solution for the given differential equation.

Now plot the graph $y = s i n 3 t + 2 c o s 3 t$ and from the graph, one can see the maximum value $|y| = 2.236$ .

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Verify that $y = s i n 3 t + 2 c o s 3 t$ is a solution to the initial value problem
$2 y'' + 18 y = 0; y (0) = 2, y' (0) = 3$
Find the maximum of $| y (t) |$ for $- \infty < t < \infty$ .

Short Answer

Step by step solution

Differentiate the value of y

Verify boundary condition.

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Verify that y=sin3t+2cos3tis a solution to the initial value problem2y''+18y=0;y(0)=2,y'(0)=3Find the maximum of|yt|for-∞<t<∞.

Short Answer

Step by step solution

Differentiate the value of y

Verify boundary condition.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

Mechanics Maths

Calculus

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Verify that $y = s i n 3 t + 2 c o s 3 t$ is a solution to the initial value problem
$2 y'' + 18 y = 0; y (0) = 2, y' (0) = 3$
Find the maximum of $| y (t) |$ for $- \infty < t < \infty$ .