Internal problem ID [3147]
Internal file name [OUTPUT/2639_Sunday_June_05_2022_08_37_52_AM_99009684/index.tex]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 2.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime }-\frac {x^{3} {\mathrm e}^{x^{2}}}{y \ln \left (y\right )}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {x^{3} {\mathrm e}^{x^{2}}}{y \ln \left (y \right )} \end {align*}
Where \(f(x)=x^{3} {\mathrm e}^{x^{2}}\) and \(g(y)=\frac {1}{\ln \left (y \right ) y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{\ln \left (y \right ) y}} \,dy &= x^{3} {\mathrm e}^{x^{2}} \,d x \\ \int { \frac {1}{\frac {1}{\ln \left (y \right ) y}} \,dy} &= \int {x^{3} {\mathrm e}^{x^{2}} \,d x} \\ \frac {y^{2} \ln \left (y \right )}{2}-\frac {y^{2}}{4}&=\frac {\left (x^{2}-1\right ) {\mathrm e}^{x^{2}}}{2}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= {\mathrm e}^{\frac {\operatorname {LambertW}\left (2 \left (x^{2} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}}+2 c_{1} \right ) {\mathrm e}^{-1}\right )}{2}+\frac {1}{2}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\frac {\operatorname {LambertW}\left (2 \left (x^{2} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}}+2 c_{1} \right ) {\mathrm e}^{-1}\right )}{2}+\frac {1}{2}} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\frac {\operatorname {LambertW}\left (2 \left (x^{2} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}}+2 c_{1} \right ) {\mathrm e}^{-1}\right )}{2}+\frac {1}{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\frac {x^{3} {\mathrm e}^{x^{2}}}{y \ln \left (y\right )}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{3} {\mathrm e}^{x^{2}}}{y \ln \left (y\right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime } y \ln \left (y\right )=x^{3} {\mathrm e}^{x^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime } y \ln \left (y\right )d x =\int x^{3} {\mathrm e}^{x^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {y^{2} \ln \left (y\right )}{2}-\frac {y^{2}}{4}=\frac {\left (x^{2}-1\right ) {\mathrm e}^{x^{2}}}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y={\mathrm e}^{\frac {\mathit {LambertW}\left (\frac {2 \left (x^{2} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}}+2 c_{1} \right )}{{\mathrm e}}\right )}{2}+\frac {1}{2}} \end {array} \]
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: 0.031 (sec). Leaf size: 54
dsolve(diff(y(x),x)=(x^3*exp(x^2))/(y(x)*ln(y(x))),y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {2}\, \sqrt {\frac {{\mathrm e}^{x^{2}} x^{2}-{\mathrm e}^{x^{2}}+2 c_{1}}{\operatorname {LambertW}\left (2 \left ({\mathrm e}^{x^{2}} x^{2}-{\mathrm e}^{x^{2}}+2 c_{1} \right ) {\mathrm e}^{-1}\right )}} \]
✓ Solution by Mathematica
Time used: 60.191 (sec). Leaf size: 106
DSolve[y'[x]==(x^3*Exp[x^2])/(y[x]*Log[y[x]]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {2 e^{x^2} \left (x^2-1\right )+4 c_1}}{\sqrt {W\left (\frac {2 e^{x^2} \left (x^2-1\right )+4 c_1}{e}\right )}} \\ y(x)\to \frac {\sqrt {2 e^{x^2} \left (x^2-1\right )+4 c_1}}{\sqrt {W\left (\frac {2 e^{x^2} \left (x^2-1\right )+4 c_1}{e}\right )}} \\ \end{align*}