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Magazine Problem 1
Use Laplace transforms to solve each of the initial-value problems in Exercises \(1-18\) : \(\frac{d y}{d t}-y=e^{3}, \quad y(0)=2\).
Problem 1
In each of the following exercises, use the Laplace transform to find the solution of the given linear system that satisfies the given initial conditions. \(\frac{d x}{d t}+y=3 e^{2 x}\), \(\frac{d y}{d t}+x=0\) \(x(0)=2, \quad y(0)=0\)
Problem 1
In each of Exercises \(1-6\) find \(\mathscr{L}^{-1}\\{H(s)\\}\) using the convolution and Table \(9.1 .\) \(H(s)=\frac{1}{s^{2}+5 s+6}\).
Problem 2
In each of the following exercises, use the Laplace transform to find the solution of the given linear system that satisfies the given initial conditions. \(\frac{d x}{d t}-2 y=0\), \(\frac{d y}{d t}+x-3 y=2\) \(x(0)=3, \quad y(0)=0 .\)
Problem 2
In each of Exercises \(1-6\) find \(\mathscr{L}^{-1}\\{H(s)\\}\) using the convolution and Table \(9.1 .\) \(H(s)=\frac{1}{s^{2}+3 s-4}\).
Problem 2
Use Laplace transforms to solve each of the initial-value problems in Exercises \(1-18\) : \(\frac{d y}{d t}+y=2 \sin t, \quad y(0)=-1\).
Problem 3
In each of Exercises \(1-6\) find \(\mathscr{L}^{-1}\\{H(s)\\}\) using the convolution and Table \(9.1 .\) \(H(s)=\frac{1}{s\left(s^{2}+9\right)}\).
Problem 3
In each of the following exercises, use the Laplace transform to find the solution of the given linear system that satisfies the given initial conditions. \(\frac{d x}{d t}-5 x+2 y=3 e^{4 t}\). \(\frac{d y}{d t}-4 x+y=0\) \(x(0)=3, \quad y(0)=0\)
Problem 3
Use Laplace transforms to solve each of the initial-value problems in Exercises \(1-18\) : \(\frac{d^{2} y}{d t^{2}}-5 \frac{d y}{d t}+6 y=0\) \(y(0)=1, \quad y^{\prime}(0)=2\)
Problem 3
Find \(\mathscr{L}\\{f(t)\\}\) for each of the functions \(f\) defined in
Exercises \(1-18\).
\(f(t)=\left\\{\begin{array}{ll}4, & 0
