Internal problem ID [8293]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 713.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {-b y a +b^{2}+a b +b^{2} x -b a \sqrt {x}-a^{2}}{a \left (-a y+b +a +b x -a \sqrt {x}\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.485 (sec). Leaf size: 116
dsolve(diff(y(x),x) = (-b*y(x)*a+b^2+a*b+b^2*x-b*a*x^(1/2)-a^2)/a/(-a*y(x)+b+a+b*x-a*x^(1/2)),y(x), singsol=all)
\[ y \relax (x ) = \frac {\RootOf \left (-x^{\frac {3}{2}} a b +b^{2} x^{2}-a^{2} \sqrt {x}-b a \sqrt {x}-2 a^{2} x +2 b x a +2 b^{2} x +a^{2}+2 a b +b^{2}+{\mathrm e}^{\RootOf \left (9 x \left (\tanh ^{2}\left (-\frac {3 \textit {\_Z}}{2}+\frac {c_{1}}{2}\right )\right ) a^{2}-9 a^{2} x +4 \,{\mathrm e}^{\textit {\_Z}}\right )}+\left (a \sqrt {x}-2 b x -2 a -2 b \right ) \textit {\_Z} +\textit {\_Z}^{2}\right )}{a} \]
✓ Solution by Mathematica
Time used: 142.821 (sec). Leaf size: 607
DSolve[y'[x] == (-a^2 + a*b + b^2 - a*b*Sqrt[x] + b^2*x - a*b*y[x])/(a*(a + b - a*Sqrt[x] + b*x - a*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a \left (a \left (-\sqrt {x}\right )+a+b x+b\right )+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,1\right ]}}{a^2} \\ y(x)\to \frac {a \left (a \left (-\sqrt {x}\right )+a+b x+b\right )+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,2\right ]}}{a^2} \\ y(x)\to \frac {a \left (a \left (-\sqrt {x}\right )+a+b x+b\right )+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,3\right ]}}{a^2} \\ y(x)\to \frac {a \left (a \left (-\sqrt {x}\right )+a+b x+b\right )+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,4\right ]}}{a^2} \\ y(x)\to \frac {a \left (a \left (-\sqrt {x}\right )+a+b x+b\right )+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,5\right ]}}{a^2} \\ y(x)\to \frac {a \left (a \left (-\sqrt {x}\right )+a+b x+b\right )+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,6\right ]}}{a^2} \\ \end{align*}