PROBLEM 3: LAPLACE TRANSFORMS OF DIFFERENTIAL EQUATIONS Find Laplace transforms of the following differential equations: a) y(t)+5y(t)-0 y(0)=2 b)2 +)0 y(o)- A: y(0)- B
5) Solve the following equation for f(t), t> 0, using Laplace transforms.
5) Solve the following equation for f(t), t> 0, using Laplace transforms.
Question (2): Laplace Transformsa) Find the Laplace Transform of the following using the Laplace Transform table provided in the back:$$ f(t)=\frac{1}{4}\left(3 e^{-2 t}-8 e^{-4 t}+9 e^{-6 t}\right) u(t) $$b) Find the inverse Laplace Transform \(F(s)\) of the following function \(f(t)\) using the table:$$ f(t)=\frac{12 s^{2}(s+1)}{\left(8 s^{2}+5 s+800\right)(s+5)^{2}(10 s+8)} $$
3. Find the solution to the following differential equation by using Laplace Transforms
6. Solve an ODE Using Laplace Transforms: For this problem you are to use Laplace Transforms. Find the complete solution for the initial value problem yº+w2y = t +u.(t - Ttcost, y(0) = 1, y(0) = 0. Hint: Look carefully at the second forcing term and rewrite cost. You can solve this by brute force using the integral below. It would be a good exercise to make sure both approaches give the same Laplace transform. The integral The solution ſeat...
Find the Laplace transforms of the following functions:
a) f(t) = sin(at + b) Using the integral of the
Laplace transform
b) f(t) = cos(t) + sin(t/2) You can directly use
table 5.1
Tableau 5.1 Transformées de Laplace les plus couramment utilisées f(t)= £. {F()} F(s)= £{f(t)} f(t)=1 F(s) = 2 f(t)=1 F(s) == 2 3 Sl)=12 F(s) n! 4 St=1" F(s)=- 5 () at F(s)- S-a n! 6 S()=1"ar F($)= (s-a)"+1 a 7 s(t)= sin(at) F(s) s? +a? S...
use laplace transforms to find ivp x"(t) + x(t) = g(t), x(0) = 3, x'(0) = -7
1. Find the Laplace transforms of these functions: r(t) = tu(t), that is, the ramp function; Ae-atu(t); Be atu(t). 2. Determine the Laplace transform of f(t) = 50cos ot u(t). 3. Obtain the Laplace transform of f(t) = (cos (2t) + e 41) u(t). 4. Find the Laplace transform of u(t-2). 5. Find vo(t) in the circuit shown below, assuming zero initial conditions. IH F + 10u(i) 42 v. (1)
LUS Wstarm onvolution. We have The CU (a) Using Laplace transforms and the convolution property, find the general slution of the initial value problem f(t), (0) 3rdr sing Laplace trans forms. (b) Solve the equation using integrating factors, Apd compare with part (a).
LUS Wstarm onvolution. We have The CU (a) Using Laplace transforms and the convolution property, find the general slution of the initial value problem f(t), (0) 3rdr sing Laplace trans forms. (b) Solve the equation using integrating...
Find the complete time-domain solution x(t) for the following differential equations using Laplace transforms. Which solutions exhibit oscillatory behavior? Which solutions exhibit convergent behavior?