60.2.355 problem 932

Internal problem ID [10942]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 932
Date solved : Tuesday, January 28, 2025 at 05:31:31 PM
CAS classification : [[_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {\left (27 y^{3}+27 \,{\mathrm e}^{3 x^{2}} y+18 \,{\mathrm e}^{3 x^{2}} y^{2}+3 y^{3} {\mathrm e}^{3 x^{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}}+27 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y+9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} y^{2}+{\mathrm e}^{\frac {9 x^{2}}{2}} y^{3}\right ) {\mathrm e}^{3 x^{2}} x \,{\mathrm e}^{-\frac {9 x^{2}}{2}}}{243 y} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 54

dsolve(diff(y(x),x) = 1/243*(27*y(x)^3+27*exp(3*x^2)*y(x)+18*exp(3*x^2)*y(x)^2+3*y(x)^3*exp(3*x^2)+27*exp(9/2*x^2)+27*exp(9/2*x^2)*y(x)+9*exp(9/2*x^2)*y(x)^2+exp(9/2*x^2)*y(x)^3)*exp(3*x^2)*x/y(x)/exp(9/2*x^2),y(x), singsol=all)
 

\[ y = \frac {369 \,{\mathrm e}^{\frac {3 x^{2}}{2}}}{-123-123 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+136 \operatorname {RootOf}\left (-41 x^{2}-50243409 \left (\int _{}^{\textit {\_Z}}\frac {1}{9248 \textit {\_a}^{3}-1860867 \textit {\_a} +1860867}d \textit {\_a} \right )+27 c_{1} \right )} \]

✓ Solution by Mathematica

Time used: 0.732 (sec). Leaf size: 129

DSolve[D[y[x],x] == (x*(27*E^((9*x^2)/2) + 27*E^(3*x^2)*y[x] + 27*E^((9*x^2)/2)*y[x] + 18*E^(3*x^2)*y[x]^2 + 9*E^((9*x^2)/2)*y[x]^2 + 27*y[x]^3 + 3*E^(3*x^2)*y[x]^3 + E^((9*x^2)/2)*y[x]^3))/(243*E^((3*x^2)/2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{-\frac {e^{3 x^2} x \left (\left (1+e^{\frac {3 x^2}{2}}\right ) y(x)+3 e^{\frac {3 x^2}{2}}\right )}{2 \sqrt [3]{34} \sqrt [3]{-e^{9 x^2} x^3} y(x)}}\frac {68}{68 K[1]^3+123 \sqrt [3]{-34} K[1]+68}dK[1]+\frac {2}{81} 34^{2/3} e^{-6 x^2} \left (-e^{9 x^2} x^3\right )^{2/3}=c_1,y(x)\right ] \]