Internal problem ID [8306]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 726.
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 {a y b -b c +b^{2} x +b a \sqrt {x}-a^{2}}{a \left (a y-c +x b +a \sqrt {x}\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.532 (sec). Leaf size: 93
dsolve(diff(y(x),x) = -(b*y(x)*a-b*c+b^2*x+b*a*x^(1/2)-a^2)/a/(a*y(x)-c+b*x+a*x^(1/2)),y(x), singsol=all)
\[ y \relax (x ) = \frac {\RootOf \left (x^{\frac {3}{2}} a b +b^{2} x^{2}-\sqrt {x}\, a c -2 a^{2} x -2 b c x +c^{2}-{\mathrm e}^{\RootOf \left (9 \left (\tanh ^{2}\left (-\frac {3 \textit {\_Z}}{2}+\frac {c_{1}}{2}\right )\right ) x \,a^{2}-9 a^{2} x -4 \,{\mathrm e}^{\textit {\_Z}}\right )}+\left (a \sqrt {x}+2 b x -2 c \right ) \textit {\_Z} +\textit {\_Z}^{2}\right )}{a} \]
✓ Solution by Mathematica
Time used: 60.076 (sec). Leaf size: 625
DSolve[y'[x] == (a^2 + b*c - a*b*Sqrt[x] - b^2*x - a*b*y[x])/(a*(-c + a*Sqrt[x] + b*x + a*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,1\right ]} \\ y(x)\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,2\right ]} \\ y(x)\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,3\right ]} \\ y(x)\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,4\right ]} \\ y(x)\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,5\right ]} \\ y(x)\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,6\right ]} \\ \end{align*}