91Ó°ÊÓ

Solve the differential equations in Exercises \(1-14\) $$ e^{2 x} y^{\prime}+2 e^{2 x} y=2 x $$

In Exercises \(5-10,\) find the orthogonal trajectories of the family of curves. Sketch sketch skeveral members of each family. $$2 x^{2}+y^{2}=c^{2}$$

a. Identify the equilibrium values. Which are stable and which are unstable? b. Construct a phase line. Identify the signs of \(y^{\prime}\) and \(y^{\prime \prime}\) . c. Sketch several solution curves. \(y^{\prime}=y^{3}-y^{2}\)

In Exercises \(7-10\) , write an equivalent first-order differential equation and initial condition for \(y .\) $$ y=\int_{1}^{x} \frac{1}{t} d t $$

In Exercises \(5-10,\) find the orthogonal trajectories of the family of curves. Sketch sketch skeveral members of each family. $$y=c e^{-x}$$

The autonomous differential equations represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0) .\) Which equilibria are stable, and which are unstable? \(\frac{d P}{d t}=1-2 P\)

Solve the differential equations in Exercises \(1-14\) $$ x y^{\prime}-y=2 x \ln x $$

In Exercises \(7-10\) , write an equivalent first-order differential equation and initial condition for \(y .\) $$ y=2-\int_{0}^{x}(1+y(t)) \sin t d t $$

The autonomous differential equations represent models for population growth. For each exercise, use a phase line analysis to sketch solution curves for \(P(t),\) selecting different starting values \(P(0) .\) Which equilibria are stable, and which are unstable? \(\frac{d P}{d t}=P(1-2 P)\)

In Exercises \(5-10,\) find the orthogonal trajectories of the family of curves. Sketch sketch skeveral members of each family. $$y=e^{k x}$$