Internal problem ID [13436]
Internal file name [OUTPUT/12608_Thursday_February_15_2024_12_01_20_AM_58626473/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 8. Review exercises for part of part II. page 143
Problem number: 15.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {\left (y^{2}-4\right ) y^{\prime }-y=0} \]
Integrating both sides gives \begin {align*} \int \frac {y^{2}-4}{y}d y &= x +c_{1}\\ \frac {y^{2}}{2}-4 \ln \left (y \right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-\frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {x}{2}-\frac {c_{1}}{2}}}{4}\right )}{2}-\frac {x}{4}-\frac {c_{1}}{4}} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{-\frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {x}{2}-\frac {c_{1}}{2}}}{4}\right )}{2}-\frac {x}{4}-\frac {c_{1}}{4}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (y^{2}-4\right ) y^{\prime }-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y}{y^{2}-4} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } \left (y^{2}-4\right )}{y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } \left (y^{2}-4\right )}{y}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2}}{2}-4 \ln \left (y\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{-\frac {\mathit {LambertW}\left (-\frac {{\mathrm e}^{-\frac {x}{2}-\frac {c_{1}}{2}}}{4}\right )}{2}-\frac {x}{4}-\frac {c_{1}}{4}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 39
dsolve((y(x)^2-4)*diff(y(x),x)=y(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 \,{\mathrm e}^{-\frac {c_{1}}{4}-\frac {x}{4}}}{\sqrt {-\frac {{\mathrm e}^{-\frac {c_{1}}{2}-\frac {x}{2}}}{\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {c_{1}}{2}-\frac {x}{2}}}{4}\right )}}} \]
✓ Solution by Mathematica
Time used: 32.653 (sec). Leaf size: 246
DSolve[(y[x]^2-4)*y'[x]==y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -2 i \sqrt {W\left (-\frac {1}{4} \sqrt [4]{e^{-2 (x+c_1)}}\right )} \\ y(x)\to 2 i \sqrt {W\left (-\frac {1}{4} \sqrt [4]{e^{-2 (x+c_1)}}\right )} \\ y(x)\to -2 i \sqrt {W\left (-\frac {1}{4} i \sqrt [4]{e^{-2 (x+c_1)}}\right )} \\ y(x)\to 2 i \sqrt {W\left (-\frac {1}{4} i \sqrt [4]{e^{-2 (x+c_1)}}\right )} \\ y(x)\to -2 i \sqrt {W\left (\frac {1}{4} i \sqrt [4]{e^{-2 (x+c_1)}}\right )} \\ y(x)\to 2 i \sqrt {W\left (\frac {1}{4} i \sqrt [4]{e^{-2 (x+c_1)}}\right )} \\ y(x)\to -2 i \sqrt {W\left (\frac {1}{4} \sqrt [4]{e^{-2 (x+c_1)}}\right )} \\ y(x)\to 2 i \sqrt {W\left (\frac {1}{4} \sqrt [4]{e^{-2 (x+c_1)}}\right )} \\ y(x)\to 0 \\ \end{align*}