91Ó°ÊÓ

State whether the equation is ordinary or partial, linear or nonlinear, and give its order. $$ \left(x^{2}+y^{2}\right) d x+2 x y d y=0 $$

For each of the following differential equations, draw several isoclines with appropriate direction markers, and sketch several solution curves for the equation. $$ \frac{d y}{d x}=\frac{2 y}{x} $$

Identify the independent variables, the dependent variables, and the parameters in the equations given as examples in this section. $$\frac{\partial u}{\partial t}=h^{2}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right)$$

State whether the equation is ordinary or partial, linear or nonlinear, and give its order. $$ y^{\prime}+P(x) y=Q(x) $$

For each of the following differential equations, draw several isoclines with appropriate direction markers, and sketch several solution curves for the equation. $$ \frac{d y}{d x}=y-x $$

Solve the differential equations. $$ \frac{d y}{d x}=\frac{4}{x^{2}-1} $$

State whether the equation is ordinary or partial, linear or nonlinear, and give its order. $$ y^{\prime \prime \prime}-3 y^{\prime}+2 y=0 $$

For each of the following differential equations, draw several isoclines with appropriate direction markers, and sketch several solution curves for the equation. $$ \frac{d y}{d x}=x+y+1 $$

Identify the independent variables, the dependent variables, and the parameters in the equations given as examples in this section. $$L \frac{d^{2} i}{d t^{2}}+R \frac{d i}{d t}+\frac{1}{C} i=E \omega \cos \omega t$$

Solve the differential equations. $$ \frac{d y}{d x}=\frac{2}{x^{2}+4} $$