Studdy AI Logo
Math  /  Calculus

Question ( 417 mintiat) hn  shen x(y′′(t))= hn  b) x2Y(x)+3x−2\begin{array}{l} \text { shen } x\left(y^{\prime \prime}(t)\right)= \\ \text { hn } \\ \text { b) } x^{2} Y(x)+3 x-2 \end{array} ch x2×(4)−26+3x^{2} \times(4)-26+3 4)3x(x)+2x2−34)^{3} x(x)+2 x^{2}-3

Studdy Solution

STEP 1

1. We are given initial conditions y(0)=−3 y(0) = -3 and y′(0)=2 y'(0) = 2 .
2. We need to find the Laplace Transform of the second derivative of y(t) y(t) , denoted as L(y′′(t)) L(y''(t)) .
3. The Laplace Transform of the second derivative y′′(t) y''(t) is given by the formula L(y′′(t))=s2Y(s)−sy(0)−y′(0) L(y''(t)) = s^2 Y(s) - s y(0) - y'(0) , where Y(s) Y(s) is the Laplace Transform of y(t) y(t) .

STEP 2

1. Write the formula for the Laplace Transform of y′′(t) y''(t) .
2. Substitute the given initial conditions into the formula.
3. Simplify the expression to match one of the provided answer choices.

STEP 3

Write the formula for the Laplace Transform of y′′(t) y''(t) .
L(y′′(t))=s2Y(s)−sy(0)−y′(0) L(y''(t)) = s^2 Y(s) - s y(0) - y'(0)

STEP 4

Substitute the given initial conditions y(0)=−3 y(0) = -3 and y′(0)=2 y'(0) = 2 into the formula.
L(y′′(t))=s2Y(s)−s(−3)−2 L(y''(t)) = s^2 Y(s) - s(-3) - 2

STEP 5

Simplify the expression.
L(y′′(t))=s2Y(s)+3s−2 L(y''(t)) = s^2 Y(s) + 3s - 2

STEP 6

Match the simplified expression with the provided answer choices. The correct answer is:
b) s2Y(s)+3s−2 \text{b) } s^2 Y(s) + 3s - 2

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord