60.2.41 problem 617
Internal
problem
ID
[10628]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
617
Date
solved
:
Tuesday, January 28, 2025 at 05:00:15 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\begin{align*} y^{\prime }&=\frac {F \left (\frac {\left (3+y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 y}\right ) x y^{2} {\mathrm e}^{3 x^{2}} {\mathrm e}^{-\frac {9 x^{2}}{2}}}{9} \end{align*}
✓ Solution by Maple
Time used: 0.110 (sec). Leaf size: 92
dsolve(diff(y(x),x) = 1/9*F(1/3*(3+y(x))*exp(3/2*x^2)/y(x))*x*y(x)^2*exp(3*x^2)/exp(9/2*x^2),y(x), singsol=all)
\begin{align*}
y &= \operatorname {RootOf}\left (F \left (\frac {\left (\textit {\_Z} +3\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}}{3 \textit {\_Z}}\right ) \textit {\_Z} \,{\mathrm e}^{3 x^{2}}-9 \,{\mathrm e}^{\frac {9 x^{2}}{2}} \textit {\_Z} -27 \,{\mathrm e}^{\frac {9 x^{2}}{2}}\right ) \\
y &= -\frac {3 \,{\mathrm e}^{\frac {3 x^{2}}{2}}}{{\mathrm e}^{\frac {3 x^{2}}{2}}-3 \operatorname {RootOf}\left (-x^{2}-18 \left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )-27 \textit {\_a}}d \textit {\_a} \right )+2 c_{1} \right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.628 (sec). Leaf size: 615
DSolve[D[y[x],x] == (x*F[(E^((3*x^2)/2)*(3 + y[x]))/(3*y[x])]*y[x]^2)/(9*E^((3*x^2)/2)),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (\frac {9 e^{\frac {3 x^2}{2}}-F\left (\frac {e^{\frac {3 x^2}{2}} (K[2]+3)}{3 K[2]}\right )}{3 \left (\left (9 e^{\frac {3 x^2}{2}}-F\left (\frac {e^{\frac {3 x^2}{2}} (K[2]+3)}{3 K[2]}\right )\right ) K[2]+27 e^{\frac {3 x^2}{2}}\right )}-\int _1^x\left (-\frac {K[2] \left (\frac {e^{\frac {3 K[1]^2}{2}}}{3 K[2]}-\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]^2}\right ) F''\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[1]}{-9 e^{\frac {3 K[1]^2}{2}} K[2]+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2]-27 e^{\frac {3 K[1]^2}{2}}}+\frac {F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2] \left (F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right )-9 e^{\frac {3 K[1]^2}{2}}+K[2] \left (\frac {e^{\frac {3 K[1]^2}{2}}}{3 K[2]}-\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]^2}\right ) F''\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right )\right ) K[1]}{\left (-9 e^{\frac {3 K[1]^2}{2}} K[2]+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2]-27 e^{\frac {3 K[1]^2}{2}}\right )^2}-\frac {F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[1]}{-9 e^{\frac {3 K[1]^2}{2}} K[2]+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (K[2]+3)}{3 K[2]}\right ) K[2]-27 e^{\frac {3 K[1]^2}{2}}}\right )dK[1]-\frac {1}{3 K[2]}\right )dK[2]+\int _1^x-\frac {F\left (\frac {e^{\frac {3 K[1]^2}{2}} (y(x)+3)}{3 y(x)}\right ) K[1] y(x)}{-9 e^{\frac {3 K[1]^2}{2}} y(x)+F\left (\frac {e^{\frac {3 K[1]^2}{2}} (y(x)+3)}{3 y(x)}\right ) y(x)-27 e^{\frac {3 K[1]^2}{2}}}dK[1]=c_1,y(x)\right ]
\]