91Ó°ÊÓ

Each \(D E\) in Problems 23-30 is of the form given in (5). In Problems 23-28, solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\frac{1-x-y}{x+y} $$

Solve the given initial-value problem. $$ \left(\frac{3 y^{2}-t^{2}}{y^{5}}\right) \frac{d y}{d t}+\frac{t}{2 y^{4}}=0, \quad y(1)=1 $$

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the \(x y\) -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=10+3 y-y^{2} $$

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the \(x y\) -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=y^{2}\left(4-y^{2}\right) $$

In Problems 23-28, find an implicit and an explicit solution of the given initial-value problem. $$ x^{2} \frac{d y}{d x}=y-x y, \quad y(-1)=-1 $$

Solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\tan ^{2}(x+y) $$

Solve the given initial-value problem. $$ \begin{aligned} &\left(y^{2} \cos x-3 x^{2} y-2 x\right) d x+\left(2 y \sin x-x^{3}+\ln y\right) d y=0 \\ &y(0)=e \end{aligned} $$

Solve the given initial-value problem. Give the largest interval \(I\) over which the solution is defined. $$ x y^{\prime}+y=e^{x}, \quad y(1)=2 $$

In Problems, find an implicit and an explicit solution of the given initial- value problem. \(x^{2} \frac{d y}{d x}=y-x y, \quad y(-1)=-1\)

Each \(D E\) in Problems 23-30 is of the form given in (5). In Problems 23-28, solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\tan ^{2}(x+y) $$