91Ó°ÊÓ

Identify the independent variables, the dependent variables, and the parameters in the equations given as examples in this section. $$\frac{d y}{d x}=\cos x$$

In Exercises 1 through \(16,\) state whether the equation is ordinary or partial, linear or nonlinear, and give its order. $$ \frac{d^{2} x}{d t^{2}}+k^{2} x=0 $$

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

Solve the differential equations in Exercises 1 through 6. $$ \frac{d y}{d x}=x^{3}+2 x $$

Consider the problem of drawing isoclines for the equation \(d y / d x=\) \(y \sin (y+x) .\) This is a very hard problem. Now use the computer to draw the slope field and representative solution curves.

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

Solve the differential equations in Exercises 1 through 6. $$ \frac{d y}{d x}=\frac{3}{x} $$

State whether the equation is ordinary or partial, linear or nonlinear, and give its order. $$ \frac{\partial^{2} w}{\partial t^{2}}=a^{2} \frac{\partial^{2} w}{\partial x^{2}} $$

Identify the independent variables, the dependent variables, and the parameters in the equations given as examples in this section. $$\frac{d^{2} y}{d x^{2}}+k^{2} y=0$$

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