54.2.45 problem 621
Internal
problem
ID
[11919]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
621
Date
solved
:
Tuesday, September 30, 2025 at 11:40:06 PM
CAS
classification
:
[[_homogeneous, `class G`], [_Abel, `2nd type`, `class C`]]
\begin{align*} y^{\prime }&=\frac {1}{y+\sqrt {x}} \end{align*}
✓ Maple. Time used: 0.025 (sec). Leaf size: 246
ode:=diff(y(x),x) = 1/(y(x)+x^(1/2));
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (4 c_1 -8 x^{{3}/{2}}+4 \sqrt {c_1 \left (c_1 -4 x^{{3}/{2}}\right )}\right )^{{2}/{3}}+4 x}{2 \left (4 c_1 -8 x^{{3}/{2}}+4 \sqrt {c_1 \left (c_1 -4 x^{{3}/{2}}\right )}\right )^{{1}/{3}}} \\
y &= \frac {i \left (-\left (4 c_1 -8 x^{{3}/{2}}+4 \sqrt {c_1 \left (c_1 -4 x^{{3}/{2}}\right )}\right )^{{2}/{3}}+4 x \right ) \sqrt {3}-\left (4 c_1 -8 x^{{3}/{2}}+4 \sqrt {c_1 \left (c_1 -4 x^{{3}/{2}}\right )}\right )^{{2}/{3}}-4 x}{4 \left (4 c_1 -8 x^{{3}/{2}}+4 \sqrt {c_1 \left (c_1 -4 x^{{3}/{2}}\right )}\right )^{{1}/{3}}} \\
y &= \frac {i \left (\left (4 c_1 -8 x^{{3}/{2}}+4 \sqrt {c_1 \left (c_1 -4 x^{{3}/{2}}\right )}\right )^{{2}/{3}}-4 x \right ) \sqrt {3}-\left (4 c_1 -8 x^{{3}/{2}}+4 \sqrt {c_1 \left (c_1 -4 x^{{3}/{2}}\right )}\right )^{{2}/{3}}-4 x}{4 \left (4 c_1 -8 x^{{3}/{2}}+4 \sqrt {c_1 \left (c_1 -4 x^{{3}/{2}}\right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 60.029 (sec). Leaf size: 445
ode=D[y[x],x] == (Sqrt[x] + y[x])^(-1);
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - 1/(sqrt(x) + y(x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - 1/(sqrt(x) + y(x)) cannot be solved by the