91Ó°ÊÓ

Find the inverse transforms of the given functions of \(s\) $$F(s)=\frac{1}{s^{3}+3 s^{2}+3 s+1}$$

Show that the given equation is a solution of the given differential equation. $$x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=0, \quad y=c_{1} \ln x+c_{2}$$

Solve the given differential equations. $$2 D^{2} y+5 y=4 D y$$

Solve the given differential equations by Laplace transforms. The function is subject to the given conditions. $$y^{\prime \prime}-4 y=10 e^{3 t}, y(0)=5, y^{\prime}(0)=0$$

Find the particular solutions to the given differential equations that satisfy the given conditions. $$y d x-x d y=y^{3} d x+y^{2} x d y ; \quad x=2 \text { when } y=4$$

Solve the given differential equations. $$D^{2} y-4 y=\sin x+2 \cos x$$

Solve the given differential equations. $$y^{\prime}=x^{3}(1-4 y)$$

Solve the given differential equations. $$y^{\prime \prime}=3 y^{\prime}+y$$

Solve the given problems by solving the appropriate differential equation. Assuming that the natural environment of Earth is limited and that the maximum population it can sustain is \(M,\) the rate of growth of the population \(P\) is given by the logistic differential equation \(\frac{d P}{d t}=k P(M-P) .\) Using this equation for Earth, if \(P=7.4\) billion in \(2016, k=0.00040,\) and \(M=25\) billion, what will be the population of Earth in \(2026 ?\)

Solve the given differential equations by Laplace transforms. The function is subject to the given conditions. $$y^{\prime \prime}-2 y^{\prime}+y=e^{2 t}, y(0)=1, y^{\prime}(0)=3$$