Internal problem ID [8201]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 621.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], [_Abel, 2nd type, class C]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {1}{y+\sqrt {x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.974 (sec). Leaf size: 59
dsolve(diff(y(x),x) = 1/(y(x)+x^(1/2)),y(x), singsol=all)
\[ y \relax (x ) = \frac {\RootOf \left (\textit {\_Z}^{18} c_{1}-9 x \,\textit {\_Z}^{6}-6 \textit {\_Z}^{3} \sqrt {x}-1\right )^{3} \sqrt {x}+1}{\RootOf \left (\textit {\_Z}^{18} c_{1}-9 x \,\textit {\_Z}^{6}-6 \textit {\_Z}^{3} \sqrt {x}-1\right )^{3}} \]
✓ Solution by Mathematica
Time used: 34.996 (sec). Leaf size: 445
DSolve[y'[x] == (Sqrt[x] + y[x])^(-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {x}+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^2+8 \text {$\#$1}^3 x^{3/2}+9 \text {$\#$1}^2 x-6 \text {$\#$1} \sqrt {x}+1\&,1\right ]} \\ y(x)\to -\sqrt {x}+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^2+8 \text {$\#$1}^3 x^{3/2}+9 \text {$\#$1}^2 x-6 \text {$\#$1} \sqrt {x}+1\&,2\right ]} \\ y(x)\to -\sqrt {x}+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^2+8 \text {$\#$1}^3 x^{3/2}+9 \text {$\#$1}^2 x-6 \text {$\#$1} \sqrt {x}+1\&,3\right ]} \\ y(x)\to -\sqrt {x}+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^2+8 \text {$\#$1}^3 x^{3/2}+9 \text {$\#$1}^2 x-6 \text {$\#$1} \sqrt {x}+1\&,4\right ]} \\ y(x)\to -\sqrt {x}+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^2+8 \text {$\#$1}^3 x^{3/2}+9 \text {$\#$1}^2 x-6 \text {$\#$1} \sqrt {x}+1\&,5\right ]} \\ y(x)\to -\sqrt {x}+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-24 \text {$\#$1}^4 x^2+8 \text {$\#$1}^3 x^{3/2}+9 \text {$\#$1}^2 x-6 \text {$\#$1} \sqrt {x}+1\&,6\right ]} \\ \end{align*}