91Ó°ÊÓ

Solve the following differential equations: $$ \frac{d y}{d t}=\frac{5-t}{y^{2}} $$

Show that the function \(f(t)=\frac{3}{2} e^{t^{2}}-\frac{1}{2}\) is a solution of the differential equation \(y^{\prime}-2 t y=t\).

You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of \(\frac{d N}{d t}\) versus \(N\) in an \(N z\) -plane. (c) In the \(t N\) -plane, plot the constant solutions and place a dashed line where the concavity of certain solutions may change. (d) Sketch the solution curve corresponding to each given initial condition. $$ d N / d t=N(1-N), N(0)=.75 $$

Suppose that \(f(t)\) is a solution of the differential equation \(y^{\prime}=t y-5\) and the graph of \(f(t)\) passes through the point \((2,4)\). What is the slope of the graph at this point?

Review concepts that are important in this section. In each exercise, sketch the graph of a function with the stated properties. Domain: \(0 \leq t \leq 3 ;(0,1)\) is on the graph; the slope is always positive, and the slope becomes less positive (as \(t\) increases).

Find an integrating factor for each equation. Take \(t>0\). $$ y^{\prime}-2 y=t $$

Review concepts that are important in this section. In each exercise, sketch the graph of a function with the stated properties. Domain: \(0 \leq t \leq 4 ;(0,2)\) is on the graph; the slope is always positive, and the slope becomes more positive (as t increases).

Suppose that \(f(t)\) is a solution of \(y^{\prime}=t^{2}-y^{2}\) and the graph of \(f(t)\) passes through the point \((2,3)\). Find the slope of the graph when \(t=2\).

Find an integrating factor for each equation. Take \(t>0\). $$ y^{\prime}+t y=6 t $$

Solve the following differential equations: $$ \frac{d y}{d t}=t e^{2 y} $$