60.2.199 problem 775
Internal
problem
ID
[10786]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
775
Date
solved
:
Monday, January 27, 2025 at 09:44:20 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational]
\begin{align*} y^{\prime }&=\frac {x -y+\sqrt {y}}{x -y+\sqrt {y}+1} \end{align*}
✓ Solution by Maple
Time used: 0.050 (sec). Leaf size: 44
dsolve(diff(y(x),x) = (x-y(x)+y(x)^(1/2))/(x-y(x)+y(x)^(1/2)+1),y(x), singsol=all)
\[
-2 y^{{3}/{2}}+y^{3}+\left (-3 x -3\right ) y^{2}+\left (3 x^{2}+3 x \right ) y-x^{3}-c_{1} = 0
\]
✓ Solution by Mathematica
Time used: 10.800 (sec). Leaf size: 943
DSolve[D[y[x],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*}