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Magazine Problem 6
Solve the differential equations. $$ \frac{d y}{d x}=\frac{3}{x^{2}+x} $$
Problem 6
For each of the following differential equations, draw several isoclines with appropriate direction markers, and sketch several solution curves for the equation. $$ \frac{d y}{d x}=x-y-1 $$
Problem 6
State whether the equation is ordinary or partial, linear or nonlinear, and give its order. $$ y y^{\prime \prime}=x $$
Problem 7
Solve the initial value problem. $$ \frac{d y}{d x}=3 e^{x}, \quad y=6, \text { when } x=0 $$
Problem 7
Identify the independent variables, the dependent variables, and the parameters in the equations given as examples in this section. $$\left(\frac{d^{2} w}{d x^{2}}\right)^{3}-x y \frac{d w}{d x}+w=0$$
Problem 7
For each of the following differential equations, draw several isoclines with appropriate direction markers, and sketch several solution curves for the equation. $$ \frac{d y}{d x}=2 x-y $$
Problem 7
State whether the equation is ordinary or partial, linear or nonlinear, and give its order. $$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=0 $$
Problem 8
For each of the following differential equations, draw several isoclines with appropriate direction markers, and sketch several solution curves for the equation. $$ \frac{d y}{d x}=y-x^{2} $$
Problem 8
State whether the equation is ordinary or partial, linear or nonlinear, and give its order. $$ \frac{d^{4} y}{d x^{4}}=w(x) $$
Problem 8
Solve the initial value problem. $$ \frac{d y}{d x}=4 e^{-3 x}, \quad y=2, \text { when } x=0 $$
