7.4E: Derivatives of Laplace Transforms and Linear Models (Exercises)

In Exercises 1-3 find the Laplace transform of the function.

1. \(te^{10t}\)

2. \(t^3e^{-2t}\)

3. \(e^{6t}(1+3t-t^2)\)

In Exercises 4-8 find the inverse Laplace transform of the function.

4. \({3\over(s-7)^4}\)

5. \({2s+3\over(s-7)^4}\)

6. \({s^2-1\over(s-2)^6}\)

7. \( {s\over s^2+2s+1}\)

8. \( {s^3+2s^2-s-3\over(s+1)^4}\)

In Exercises 9-13 solve the initial value problem using Laplace transforms.

9. \(y'+4y=e^{-4t}, \quad y(0)=2,\)

10. \(y''+2y'+y=0, \quad y(0)=1,\; y'(0)=1\)

11. \(y''-6y'+9y=t, \quad y(0)=0,\; y'(0)=1\)

12. \(y''+2y'+y=6\sin t-4\cos t, \quad y(0)=-1,\; y'(0)=1\)

13. \(4y''+4y'+y=3\sin t+\cos t, \quad y(0)=2,\; y'(0)=-1\)

In Exercises 14-15, solve the application problem using Laplace transforms.

14. Suppose that you brew a cup of coffee at \(183\) degrees in a room with temperature of \(78\) degrees. Assume a constant of proportionality \(k=-0.425\). Let \(T(t)\) be the temperature function of the coffee at any time \(t>0\). Use Laplace transforms to solve the differential equation for the temperature of the coffee and determine how long it takes before the coffee reaches \(167\) degrees.

15. An object with mass \(13\) kg is attached to a spring with constant \(k=52\) kg/m and subjected to an external force \(F⁡(t)= -1365⁢\cos⁡(5⁢t)\). The object begins at rest in its equilibrium position. Use the Laplace transform to find its displacement for \(t>0\).

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