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Magazine Problem 15
By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}+9 y=\cos 3 t, \quad y_{0}=0, \quad y_{0}^{\prime}=6 $$
Problem 16
By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}+9 y=\cos 3 t, \quad y_{0}=2, \quad y_{0}^{\prime}=0 $$
Problem 17
Solve the differential equation \(y^{\prime \prime}-a^{2} y=f(t)\), where, $$ f(t)=\left\\{\begin{array}{ll} 0, & t<0 \\ 1, & t>0 \end{array} \quad \text { and } y_{0}=y_{0}^{\prime}=0\right. $$ Hint: Lse the convolution integral as in the example.
Problem 17
By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}+5 y^{\prime}+6 y=12, \quad y_{0}=2, \quad y_{0}^{\prime}=0 $$
Problem 18
A mechanical or electrical system is described by the differential equation
\(y^{n}+\omega^{2} y=f(t)\). Find \(y\) if
$$
f(t)=\left\\{\begin{array}{ll}
1, & 0
Problem 19
By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}+y^{\prime}-5 y=e^{2 t}, \quad y_{0}=1, \quad y_{0}^{\prime}=2 $$
Problem 20
By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}-8 y^{\prime}+16 y=32 t, \quad y_{0}=1, \quad y_{0}^{\prime}=2 $$
Problem 21
By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}+4 y^{\prime}+5 y=26 e^{3 t}, \quad y_{0}=1, \quad y_{0}^{\prime}=5 $$
Problem 22
By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}+2 y^{\prime}+5 y=10 \cos t, \quad y_{0}=2, \quad y_{0}^{\prime}=1 $$
Problem 24
(a) Find the exponential Fourier transform of \(f(x)=e^{-|x|}\) and write the inverse transform. You should find $$ \int_{0}^{\infty} \frac{\cos x x}{x^{2}+1} d x=\frac{\pi}{2} e^{-|x|} $$ (b) Also obtain the result in (a) by using the Fourier cosine transform equations (4.15). (c) Find the Fourier cosine transform of \(f(x)=1 /\left(1+x^{2}\right)\). Hint : Write your result in (b) with \(x\) and \(\alpha\) interchanged.
