60.2.30 problem 606

Internal problem ID [10617]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 606
Date solved : Tuesday, January 28, 2025 at 04:56:07 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=-\left (-{\mathrm e}^{-x^{2}}+x^{2} {\mathrm e}^{-x^{2}}-F \left (y-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}\right )\right ) x \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 34

dsolve(diff(y(x),x) = -(-exp(-x^2)+x^2*exp(-x^2)-F(y(x)-1/2*x^2*exp(-x^2)))*x,y(x), singsol=all)
 

\[ y = \frac {{\mathrm e}^{-x^{2}} x^{2}}{2}+\operatorname {RootOf}\left (x^{2}-2 \left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} \right )+2 c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.395 (sec). Leaf size: 361

DSolve[D[y[x],x] == x*(E^(-x^2) - x^2/E^x^2 + F[-1/2*x^2/E^x^2 + y[x]]),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (K[2]-\frac {1}{2} e^{-x^2} x^2\right ) \int _1^x\left (\frac {e^{-K[1]^2} F''\left (K[2]-\frac {1}{2} e^{-K[1]^2} K[1]^2\right ) K[1]^3}{F\left (K[2]-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )^2}-\frac {e^{-K[1]^2} \left (e^{K[1]^2} F\left (K[2]-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )+1\right ) F''\left (K[2]-\frac {1}{2} e^{-K[1]^2} K[1]^2\right ) K[1]}{F\left (K[2]-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )^2}+\frac {F''\left (K[2]-\frac {1}{2} e^{-K[1]^2} K[1]^2\right ) K[1]}{F\left (K[2]-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )}\right )dK[1]+1}{F\left (K[2]-\frac {1}{2} e^{-x^2} x^2\right )}dK[2]+\int _1^x\left (\frac {e^{-K[1]^2} \left (e^{K[1]^2} F\left (y(x)-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )+1\right ) K[1]}{F\left (y(x)-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )}-\frac {e^{-K[1]^2} K[1]^3}{F\left (y(x)-\frac {1}{2} e^{-K[1]^2} K[1]^2\right )}\right )dK[1]=c_1,y(x)\right ] \]