Internal problem ID [9279]
Internal file name [OUTPUT/8215_Monday_June_06_2022_02_19_19_AM_87185876/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 946.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class C`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left ({\mathrm e}^{-3 x^{2}} x^{6}-6 \,{\mathrm e}^{-2 x^{2}} x^{4} y-4 \,{\mathrm e}^{-2 x^{2}} x^{4}+12 x^{2} {\mathrm e}^{-x^{2}} y^{2}+8 x^{2} {\mathrm e}^{-x^{2}} y+4 x^{2} {\mathrm e}^{-2 x^{2}}+8 x^{2} {\mathrm e}^{-x^{2}}-8 y^{3}-8 \,{\mathrm e}^{-x^{2}} y-8 \,{\mathrm e}^{-x^{2}}\right ) x}{-8 y+4 x^{2} {\mathrm e}^{-x^{2}}-8}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -\left ({\mathrm e}^{-x^{2}}\right )^{3} x^{7}+6 \left ({\mathrm e}^{-x^{2}}\right )^{2} y x^{5}+4 \left ({\mathrm e}^{-x^{2}}\right )^{2} x^{5}-12 \,{\mathrm e}^{-x^{2}} y^{2} x^{3}-8 y \,{\mathrm e}^{-x^{2}} x^{3}-4 x^{3} \left ({\mathrm e}^{-x^{2}}\right )^{2}+8 x y^{3}+4 y^{\prime } {\mathrm e}^{-x^{2}} x^{2}-8 \,{\mathrm e}^{-x^{2}} x^{3}+8 y \,{\mathrm e}^{-x^{2}} x -8 y^{\prime } y+8 \,{\mathrm e}^{-x^{2}} x -8 y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left ({\mathrm e}^{-x^{2}}\right )^{3} x^{7}-6 \left ({\mathrm e}^{-x^{2}}\right )^{2} y x^{5}-4 \left ({\mathrm e}^{-x^{2}}\right )^{2} x^{5}+12 \,{\mathrm e}^{-x^{2}} y^{2} x^{3}+8 y \,{\mathrm e}^{-x^{2}} x^{3}+4 x^{3} \left ({\mathrm e}^{-x^{2}}\right )^{2}-8 x y^{3}+8 \,{\mathrm e}^{-x^{2}} x^{3}-8 y \,{\mathrm e}^{-x^{2}} x -8 \,{\mathrm e}^{-x^{2}} x}{-8 y+4 x^{2} {\mathrm e}^{-x^{2}}-8} \end {array} \]
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: 85
dsolve(diff(y(x),x) = (-8*exp(-x^2)*y(x)+4*x^2*exp(-x^2)^2-8*exp(-x^2)+8*x^2*exp(-x^2)*y(x)-4*x^4*exp(-x^2)^2+8*x^2*exp(-x^2)-8*y(x)^3+12*x^2*exp(-x^2)*y(x)^2-6*y(x)*x^4*exp(-x^2)^2+x^6*exp(-x^2)^3)*x/(-8*y(x)+4*x^2*exp(-x^2)-8),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {2+x^{2} \left (\sqrt {-x^{2}+c_{1}}-1\right ) {\mathrm e}^{-x^{2}}}{2 \sqrt {-x^{2}+c_{1}}-2} \\ y \left (x \right ) &= \frac {-2+x^{2} \left (\sqrt {-x^{2}+c_{1}}+1\right ) {\mathrm e}^{-x^{2}}}{2 \sqrt {-x^{2}+c_{1}}+2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.085 (sec). Leaf size: 93
DSolve[y'[x] == (x*(-8/E^x^2 + (4*x^2)/E^(2*x^2) + (8*x^2)/E^x^2 - (4*x^4)/E^(2*x^2) + x^6/E^(3*x^2) - (8*y[x])/E^x^2 + (8*x^2*y[x])/E^x^2 - (6*x^4*y[x])/E^(2*x^2) + (12*x^2*y[x]^2)/E^x^2 - 8*y[x]^3))/(-8 + (4*x^2)/E^x^2 - 8*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-x^2} x^2+\frac {8}{-8+\sqrt {-64 x^2+c_1}} \\ y(x)\to \frac {1}{2} e^{-x^2} x^2-\frac {8}{8+\sqrt {-64 x^2+c_1}} \\ y(x)\to \frac {1}{2} e^{-x^2} x^2 \\ \end{align*}