Internal problem ID [9052]
Internal file name [OUTPUT/7987_Monday_June_06_2022_01_08_17_AM_83877208/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 718.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[_Abel]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\left (1+y^{2} {\mathrm e}^{2 x^{2}}+y^{3} {\mathrm e}^{3 x^{2}}\right ) {\mathrm e}^{-x^{2}} x=0} \]
This is Abel first kind ODE, it has the form \[ y^{\prime }= f_0(x)+f_1(x) y +f_2(x)y^{2}+f_3(x)y^{3} \] Comparing the above to given ODE which is \begin {align*} y^{\prime }&=y^{3} {\mathrm e}^{3 x^{2}} {\mathrm e}^{-x^{2}} x +y^{2} {\mathrm e}^{2 x^{2}} {\mathrm e}^{-x^{2}} x +{\mathrm e}^{-x^{2}} x\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= {\mathrm e}^{-x^{2}} x\\ f_1(x) &= 0\\ f_2(x) &= x \,{\mathrm e}^{x^{2}}\\ f_3(x) &= x \,{\mathrm e}^{2 x^{2}} \end {align*}
Since \(f_2(x)=x \,{\mathrm e}^{x^{2}}\) is not zero, then the first step is to apply the following transformation to remove \(f_2\). Let \(y = u(x) - \frac {f_2}{3 f_3}\) or \begin {align*} y &= u(x) - \left ( \frac {x \,{\mathrm e}^{x^{2}}}{3 x \,{\mathrm e}^{2 x^{2}}} \right ) \\ &= u \left (x \right )-\frac {{\mathrm e}^{-x^{2}}}{3} \end {align*}
The above transformation applied to (1) gives a new ODE as \begin {align*} u^{\prime }\left (x \right ) = x \,{\mathrm e}^{2 x^{2}} u \left (x \right )^{3}-\frac {x u \left (x \right )}{3}+\frac {11 \,{\mathrm e}^{-x^{2}} x}{27}\tag {2} \end {align*}
This is Abel first kind ODE, it has the form \[ u^{\prime }\left (x \right )= f_0(x)+f_1(x) u \left (x \right ) +f_2(x)u \left (x \right )^{2}+f_3(x)u \left (x \right )^{3} \] Comparing the above to given ODE which is \begin {align*} u^{\prime }\left (x \right )&=x \,{\mathrm e}^{2 x^{2}} u \left (x \right )^{3}-\frac {x u \left (x \right )}{3}+\frac {11 \,{\mathrm e}^{-x^{2}} x}{27}\tag {1} \end {align*}
Therefore \begin {align*} f_0(x) &= \frac {11 \,{\mathrm e}^{-x^{2}} x}{27}\\ f_1(x) &= -\frac {x}{3}\\ f_2(x) &= 0\\ f_3(x) &= x \,{\mathrm e}^{2 x^{2}} \end {align*}
Since \(f_2(x)=0\) then we check the Abel invariant to see if it depends on \(x\) or not. The Abel invariant is given by \begin {align*} -\frac {f_{1}^{3}}{f_{0}^{2} f_{3}} \end {align*}
Which when evaluating gives \begin {align*} -\frac {531441 {\left (-\left (-\frac {22 x^{2} {\mathrm e}^{-x^{2}}}{27}+\frac {11 \,{\mathrm e}^{-x^{2}}}{27}\right ) x \,{\mathrm e}^{2 x^{2}}+\frac {11 \,{\mathrm e}^{-x^{2}} x \left ({\mathrm e}^{2 x^{2}}+4 x^{2} {\mathrm e}^{2 x^{2}}\right )}{27}-\frac {11 \,{\mathrm e}^{x^{2}} x^{3}}{27}\right )}^{3} {\mathrm e}^{-3 x^{2}}}{161051 x^{9}} \end {align*}
Since the Abel invariant depends on \(x\) then unable to solve this ode at this time.
Unable to complete the solution now.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-\left (1+y^{2} {\mathrm e}^{2 x^{2}}+y^{3} {\mathrm e}^{3 x^{2}}\right ) {\mathrm e}^{-x^{2}} x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (1+y^{2} {\mathrm e}^{2 x^{2}}+y^{3} {\mathrm e}^{3 x^{2}}\right ) {\mathrm e}^{-x^{2}} x \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 49
dsolve(diff(y(x),x) = (1+y(x)^2*exp(2*x^2)+y(x)^3*exp(3*x^2))*exp(-x^2)*x,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {11 \,{\mathrm e}^{-x^{2}} \operatorname {RootOf}\left (-5 x^{2}+20250 \left (\int _{}^{\textit {\_Z}}\frac {1}{121 \textit {\_a}^{3}+3375 \textit {\_a} -3375}d \textit {\_a} \right )+6 c_{1} \right )}{45}-\frac {{\mathrm e}^{-x^{2}}}{3} \]
✓ Solution by Mathematica
Time used: 0.249 (sec). Leaf size: 127
DSolve[y'[x] == (x*(1 + E^(2*x^2)*y[x]^2 + E^(3*x^2)*y[x]^3))/E^x^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {11}{3} \text {RootSum}\left [11 \text {$\#$1}^3+15 \sqrt [3]{11} \text {$\#$1}+11\&,\frac {\log \left (\frac {3 e^{2 x^2} x y(x)+e^{x^2} x}{\sqrt [3]{11} \sqrt [3]{e^{3 x^2} x^3}}-\text {$\#$1}\right )}{11 \text {$\#$1}^2+5 \sqrt [3]{11}}\&\right ]=\frac {11^{2/3} e^{x^2} x^3}{18 \sqrt [3]{e^{3 x^2} x^3}}+c_1,y(x)\right ] \]