Chapter 1: Introduction to Differential Equations

In Problemsstate the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with.

Determine whether the given differential equation is exact. If it is exact, solve it.

$\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{lnx}\mathbf{+}\frac{\mathbf{y}}{\mathbf{x}}\mathbf{)}\mathit{d}\mathit{x}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{lnx}\mathbf{)}\mathit{d}\mathit{y}$

In Problems 5-14solve the given linear system.

$\mathit{X}\mathbf{\text{'}}\mathbf{=}\mathbf{\left(}\begin{array}{cc}\mathbf{-}\mathbf{2}& \mathbf{5}\\ \mathbf{-}\mathbf{2}& \mathbf{4}\end{array}\mathbf{\right)}\mathit{X}$

In Problems 7–12 match each of the given differential equations with one or more of these solutions:

(a) $\mathit{y}\mathbf{=}\mathbf{0}$, (b) $\mathit{y}\mathbf{=}\mathbf{2}$, (c) $\mathit{y}\mathbf{=}\mathbf{2}\mathit{x}$, (d) $\mathit{y}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{2}}$

$\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{2}$

In Problems, 1-10, use the improved Euler's method to obtain a four-decimal approximation of the indicated value. First use $\mathbf{h}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{1}$ and then use $\mathbf{h}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{05}$.

${\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{3}\mathbf{y}\mathbf{+}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{5}\mathbf{;}\mathbf{y}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{)}$

Use the RK4 method with$\mathbf{h}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{1}$to approximate$\mathbf{y}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{)}$, where$\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$is the solution of the initial-value problem${\mathit{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{-}\mathbf{1}{\mathbf{)}}^{\mathbf{2}}\mathbf{,}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}$. Compare this approximate value with the actual value obtained in Problem 11 in Exercises 9.1.

In Problemsanddetermine whether the given first-order differential equation is linear in the indicated dependent variable by matching it with the first differential equation given in.

in; in.