A. Partial fraction expansion
B. Apply initial conditions in Laplace domain
C. Take Laplace transform of differential equation
D. Obtain time domain solution
Choose the correct answer from the options given below :
Solving Second-Order Differential Equations with Laplace Transform
Solving a second-order differential equation using the Laplace transform involves a specific sequence of steps. Here's the correct order:
Step 1: Take Laplace Transform (C)
The process begins by applying the Laplace transform to the entire differential equation. This converts the differential equation in the time domain ($t$) into an algebraic equation in the Laplace domain ($s$).
Step 2: Apply Initial Conditions (B)
After transforming the equation, substitute the given initial conditions (e.g., $y(0)$ and $y'(0)$ for a second-order equation) into the transformed equation. This step incorporates the specific starting state of the system.
Step 3: Partial Fraction Expansion (A)
Rearrange the equation obtained in the previous step to solve for $Y(s)$ (the Laplace transform of the solution $y(t)$). Often, $Y(s)$ will be a rational function. Use partial fraction expansion to break down $Y(s)$ into simpler terms that are easier to work with.
Step 4: Obtain Time Domain Solution (D)
Finally, take the inverse Laplace transform of the simplified expression for $Y(s)$ (obtained after partial fraction expansion). This converts the solution back from the Laplace domain ($s$) to the time domain ($t$), yielding the solution $y(t)$.
Conclusion
Therefore, the correct sequence for solving a second-order differential equation using the Laplace transform is C, B, A, D.
Correct sequence: C, B, A, D
Similar Questions
- If $A \in R_{n \times n}$; $\det A=0$, then :
A. A is non singular and the rows and columns of A are linearly independent
B. A is non-singular and the rows and columns of A are linearly dependent
C. A is non-singular and A has one zero rows
D. A is singular
E. A is singular and rows and columns of A are linearly dependent
Choose the correct answer from the options given below : - The matrix equation $AX=0$, represents :
- The greatest rate of increase of $f = xy^{2}z^{3}$ at the point $(0, -1, -2)$ is :
- The derivative of $y = \int_{0}^{\ln x} \sin e^{t} dt$ is :
- If $\lambda = 0$, is an eigen values of A, then $\det(A)$ is :
- If A is a null Matrix, then its rank is :
- Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : If $u = xy f\left(\frac{y}{x}\right)$, then $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 2u$
Reason (R) : Given function u is homogeneous of degree 2 in $x$ and $y$.
In the light of the above statements, choose the most appropriate answer from the options given below : - Determining Eigenvalues of a matrix :
A. Solve the characteristic equation
B. Form matrix $A - \lambda I$
C. Compute determinant
D. Identify eigenvalues
Choose the correct answer from the options given below : - A lag network for compensation normally consists of :
A. R, L and C elements
B. R and L elements
C. R and C elements
D. R only
E. L only
Choose the correct answer from the options given below : - A lead compensating network :
A. Improves response time
B. Stabilizes the system with low phase margin
C. Enables moderate increase in gain without affecting stability
D. Increases the resonant frequency
E. Decreases the resonant frequency
Choose the correct answer from the options given below :
Important Questions from Mixed Topic (CUET PG)
- Kalpsutra, the illustrated canonical text is from:-
- The Harappan city almost exclusively devoted to craft production was-:
- Mohandas Karamchand Gandhi launched quit India movement after the failure of:-
- The "Objectives Resolution" was introduced in constituent assembly by:-
- Who among the following was not a member of the constituent assembly:-
