2.239 problem 815

Internal problem ID [9794]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 815
Date solved : Thursday, October 17, 2024 at 10:07:14 PM
CAS classification : [[_Abel, `2nd type`, `class C`]]

Solve

\begin{align*} y^{\prime }&=\frac {\left (3+y\right )^{3} {\mathrm e}^{\frac {9 x^{2}}{2}} x \,{\mathrm e}^{\frac {3 x^{2}}{2}} {\mathrm e}^{-3 x^{2}}}{243 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81 \,{\mathrm e}^{\frac {3 x^{2}}{2}} y+243 y} \end{align*}

Unknown ode type.

2.239.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (y \left (x \right )+3\right )^{3} {\mathrm e}^{\frac {9 x^{2}}{2}} x \,{\mathrm e}^{\frac {3 x^{2}}{2}}}{81 \left (3 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+{\mathrm e}^{\frac {3 x^{2}}{2}} y \left (x \right )+3 y \left (x \right )\right ) {\mathrm e}^{3 x^{2}}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\left (y \left (x \right )+3\right )^{3} {\mathrm e}^{\frac {9 x^{2}}{2}} x \,{\mathrm e}^{\frac {3 x^{2}}{2}}}{81 \left (3 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+{\mathrm e}^{\frac {3 x^{2}}{2}} y \left (x \right )+3 y \left (x \right )\right ) {\mathrm e}^{3 x^{2}}} \end {array} \]

2.239.2 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`
 

2.239.3 Maple dsolve solution

Solving time : 0.009 (sec)
Leaf size : 165

dsolve(diff(y(x),x) = 1/81*(3+y(x))^3*exp(9/2*x^2)*x*exp(3/2*x^2)/(3*exp(3/2*x^2)+y(x)*exp(3/2*x^2)+3*y(x))/exp(3*x^2), 
       y(x),singsol=all)
 

\[ 5 \ln \left (3\right )-5 \ln \left (7\right )+5 \ln \left (\frac {\left (-81 y^{2}-243 y\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}+\left (3+y\right )^{2} {\mathrm e}^{3 x^{2}}-243 y^{2}}{\left ({\mathrm e}^{\frac {3 x^{2}}{2}} \left (3+y\right )+3 y\right )^{2}}\right )-\frac {30 \sqrt {93}\, \operatorname {arctanh}\left (\frac {\left (29 y \,{\mathrm e}^{\frac {3 x^{2}}{2}}+87 \,{\mathrm e}^{\frac {3 x^{2}}{2}}+81 y\right ) \sqrt {93}}{\left (279 y+837\right ) {\mathrm e}^{\frac {3 x^{2}}{2}}+837 y}\right )}{31}-10 \ln \left (\frac {{\mathrm e}^{\frac {3 x^{2}}{2}} \left (3+y\right )}{{\mathrm e}^{\frac {3 x^{2}}{2}} \left (3+y\right )+3 y}\right )+15 x^{2}-c_{1} = 0 \]

2.239.4 Mathematica DSolve solution

Solving time : 7.721 (sec)
Leaf size : 103

DSolve[{D[y[x],x] == (E^(3*x^2)*x*(3 + y[x])^3)/(81*(3*E^((3*x^2)/2) + 3*y[x] + E^((3*x^2)/2)*y[x])),{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {1}{186} \left (6 \sqrt {93} \text {arctanh}\left (\frac {81 y(x)-2 e^{\frac {3 x^2}{2}} (y(x)+3)}{9 \sqrt {93} y(x)}\right )+31 \log \left (-81 e^{\frac {3 x^2}{2}} (y(x)+3) y(x)+e^{3 x^2} (y(x)+3)^2-243 y(x)^2\right )\right )-\frac {1}{3} \log (y(x)+3)=c_1,y(x)\right ] \]