60.2.396 problem 973

Internal problem ID [10983]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 973
Date solved : Monday, January 27, 2025 at 10:38:33 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Abel]

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

✓ Solution by Maple

Time used: 0.697 (sec). Leaf size: 129

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

\[ y = \frac {\tan \left (\operatorname {RootOf}\left (\sqrt {-{\mathrm e}^{2 b x} \left (-3+4 b \right )}\, \ln \left (-\frac {\left (\tan \left (\textit {\_Z} \right )^{2}+1\right ) \left (-3+4 b \right )}{\left (\tan \left (\textit {\_Z} \right ) \sqrt {-{\mathrm e}^{2 b x} \left (-3+4 b \right )}-{\mathrm e}^{b x}\right )^{2}}\right )+\sqrt {-{\mathrm e}^{2 b x} \left (-3+4 b \right )}\, c_{1} +2 \sqrt {-{\mathrm e}^{2 b x} \left (-3+4 b \right )}\, x +2 \textit {\_Z} \,{\mathrm e}^{b x}\right )\right ) \sqrt {{\mathrm e}^{2 b x} \left (3-4 b \right )}}{2}-\frac {{\mathrm e}^{b x}}{2} \]

✓ Solution by Mathematica

Time used: 0.305 (sec). Leaf size: 100

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

\[ \text {Solve}\left [\int _1^{\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b-7) e^{-3 b x}}}}\frac {1}{K[1]^3-\frac {3 (3 b-2) K[1]}{(9 b-7)^{2/3}}+1}dK[1]=\frac {1}{9} x e^{2 b x} \left ((9 b-7) e^{-3 b x}\right )^{2/3}+c_1,y(x)\right ] \]