Internal problem ID [8355]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 775.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {x -y+\sqrt {y}}{x -y+\sqrt {y}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 47
dsolve(diff(y(x),x) = (x-y(x)+y(x)^(1/2))/(x-y(x)+y(x)^(1/2)+1),y(x), singsol=all)
\[ y \relax (x )^{3}-3 y \relax (x )^{2} x -3 y \relax (x )^{2}+3 x^{2} y \relax (x )-2 y \relax (x )^{\frac {3}{2}}+3 x y \relax (x )-x^{3}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 10.89 (sec). Leaf size: 943
DSolve[y'[x] == (x + Sqrt[y[x]] - y[x])/(1 + x + Sqrt[y[x]] - y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [\text {$\#$1}^6+\text {$\#$1}^5 (-6 x-6)+\text {$\#$1}^4 \left (15 x^2+24 x+9\right )+\text {$\#$1}^3 \left (-20 x^3-36 x^2-18 x-4+2 e^{3 c_1}\right )+\text {$\#$1}^2 \left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right )+\text {$\#$1} \left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right )+x^6-2 e^{3 c_1} x^3+e^{6 c_1}\&,1\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^6+\text {$\#$1}^5 (-6 x-6)+\text {$\#$1}^4 \left (15 x^2+24 x+9\right )+\text {$\#$1}^3 \left (-20 x^3-36 x^2-18 x-4+2 e^{3 c_1}\right )+\text {$\#$1}^2 \left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right )+\text {$\#$1} \left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right )+x^6-2 e^{3 c_1} x^3+e^{6 c_1}\&,2\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^6+\text {$\#$1}^5 (-6 x-6)+\text {$\#$1}^4 \left (15 x^2+24 x+9\right )+\text {$\#$1}^3 \left (-20 x^3-36 x^2-18 x-4+2 e^{3 c_1}\right )+\text {$\#$1}^2 \left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right )+\text {$\#$1} \left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right )+x^6-2 e^{3 c_1} x^3+e^{6 c_1}\&,3\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^6+\text {$\#$1}^5 (-6 x-6)+\text {$\#$1}^4 \left (15 x^2+24 x+9\right )+\text {$\#$1}^3 \left (-20 x^3-36 x^2-18 x-4+2 e^{3 c_1}\right )+\text {$\#$1}^2 \left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right )+\text {$\#$1} \left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right )+x^6-2 e^{3 c_1} x^3+e^{6 c_1}\&,4\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^6+\text {$\#$1}^5 (-6 x-6)+\text {$\#$1}^4 \left (15 x^2+24 x+9\right )+\text {$\#$1}^3 \left (-20 x^3-36 x^2-18 x-4+2 e^{3 c_1}\right )+\text {$\#$1}^2 \left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right )+\text {$\#$1} \left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right )+x^6-2 e^{3 c_1} x^3+e^{6 c_1}\&,5\right ] \\ y(x)\to \text {Root}\left [\text {$\#$1}^6+\text {$\#$1}^5 (-6 x-6)+\text {$\#$1}^4 \left (15 x^2+24 x+9\right )+\text {$\#$1}^3 \left (-20 x^3-36 x^2-18 x-4+2 e^{3 c_1}\right )+\text {$\#$1}^2 \left (15 x^4+24 x^3+9 x^2-6 e^{3 c_1} x-6 e^{3 c_1}\right )+\text {$\#$1} \left (-6 x^5-6 x^4+6 e^{3 c_1} x^2+6 e^{3 c_1} x\right )+x^6-2 e^{3 c_1} x^3+e^{6 c_1}\&,6\right ] \\ \end{align*}