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Initial value problem using Laplace Transform
I have to solve the IVP: $$\begin{cases} y''+by'-cy=1\\ y(0)=y_0\\ y'(0)=y'_0 \end{cases}$$
Suppose $$Y(s)=\frac{s^2+2s+1}{s^3+3s^2+2s}$$
We have to find $b,c,y_0$ and $y'_0$
I have found that: $$(s^2+s+1)Y(s)-(s+1)y_0-y'_0=1/s$$
How do I proceed further
- laplace-transform
- initial-value-problems
- inverse-laplace
2 Answers 2
According to the given ODE $y''+by'-cy=1$ , your last line should be $$(s^2+bs-c)Y(s)-(s+b)y_0-y'_0=1/s$$ which implies $$Y(s)=\frac{y_0s^2+(by_0+y'_0)s+1}{s^3+bs^2-cs}.$$ Now, by comparing it with $Y(s)=\frac{s^2+2s+1}{s^3+3s^2+2s}$ , we are able to find $b$ , $c$ , $y_0$ and $y'_0$ easily: $$b=3,c=-2, y_0=1, y'_0=-1.$$
- $\begingroup$ thank you so much, found this method easy to understand $\endgroup$ – amspsingh04 May 30 at 21:48
You have that the Laplace transform of the solution $y(t) $ is $Y(s) $ . So the inverse Laplace transform of $Y(s) $ gives us the solution $y(t) =\frac{e^{-2t}+1}{2}$ . From this we can calculate $y(0)=1$ and $y'(0) =-1$ . Now sustitute the expression of $y(t)$ in the equation and you'll get $$e^{-2t}(4-2b+c) -(2+c) =0.$$ Since $e^{-2t}$ and $1$ are linearly independent we Have that $b=3$ and $c=-2$ .
- 1 $\begingroup$ understood it, thank you so much! $\endgroup$ – amspsingh04 May 30 at 21:48
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- Calculus and Analysis
- Differential Equations
Laplace transform
Solve the initial value problem using Laplace transforms. y"+y=f(t) , y(0)=0 , y'(0)=1
Lennie Carroll
Answered question
Solve the initial value problem using Laplace transforms. y " + y = f ( t ) , y ( 0 ) = 0 , y ′ ( 0 ) = 1 Here f ( t ) = { 0 0 ≤ t < 3 π 1 t ≥ 3 π
Answer & Explanation
Arnold Odonnell
Skilled 2021-09-07 Added 109 answers
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Recalculate according to your conditions!
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Browse Course Material
Course info, instructors.
- Prof. Arthur Mattuck
- Prof. Haynes Miller
- Dr. Jeremy Orloff
- Dr. John Lewis
Departments
- Mathematics
As Taught In
- Mathematics Differential Equations Linear Algebra
Learning Resource Types
Differential equations, laplace transform: solving initial value problems.
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Session Overview
Session activities.
Read the course notes:
- Laplace Transform: Solving Initial Value Problems: Introduction (PDF)
- Laplace Transform Table (PDF)
- Table Entries: Derivative Rules (PDF)
- Precise Definition of Laplace Inverse (PDF)
- Laplace: Solving Initial Value Problems (PDF)
- IVP’s and t-translation (PDF)
- IVP’s: Longer Examples (PDF)
Watch the problem solving video:
- Laplace: Solving ODE’s
Complete the practice problems:
- Practice Problems 29 (PDF)
- Practice Problems 29 Solutions (PDF)
Check Yourself
Complete the problem sets:
Problem Set Part I Problems (PDF)
Problem Set Part I Solutions (PDF)
Problem Set Part II Problems (PDF)
Problem Set Part II Solutions (PDF)
VIDEO
COMMENTS
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This section provides materials for a session on operations on the simple relation between the Laplace transform of a function and the Laplace transform of its derivative