91Ó°ÊÓ

Find the general solution of the differential equation. \(y \ln x-x y^{\prime}=0\)

Write and solve the differential equation that models the verbal statement. The rate of change of \(Q\) with respect to \(t\) is inversely proportional to the square of \(t\).

Write and solve the differential equation that models the verbal statement. The rate of change of \(N\) with respect to \(s\) is proportional to \(250-s .\)

Slope Fields,( a) sketch an approximate solution of the differential equation satisfying the initial condition by hand on the slope field, (b) find the particular solution that satisfies the initial condition, and (c) use a graphing utility to graph the particular solution. Compare the graph with the handdrawn graph of part (a). To print an enlarged copy of the graph, select the MathGraph button. $$ \begin{array}{ll} \underline{\text { Function }} & \underline{\text { Differential Equation }} \\\ \frac{d y}{d x}=e^{x}-y& \quad (0,1) \end{array} $$

Find the function \(y=f(t)\) passing through the point \((0,10)\) with the given first derivative. Use a graphing utility to graph the solution. $$ \frac{d y}{d t}=\frac{3}{4} y $$

Write and solve the differential equation that models the verbal statement. Evaluate the solution at the specified value of the independent variable. The rate of change of \(N\) is proportional to \(N .\) When \(t=0\), \(N=250\) and when \(t=1, N=400 .\) What is the value of \(N\) when \(t=4 ?\)

Solve the Bernoulli differential equation. $$ y^{\prime}+3 x^{2} y=x^{2} y^{3} $$

Solve the Bernoulli differential equation. $$ y^{\prime}+\left(\frac{1}{x}\right) y=x \sqrt{y} $$

Solve the Bernoulli differential equation. $$ y y^{\prime}-2 y^{2}=e^{x} $$

Give the differential equation that models exponential growth and decay.