56.1.21 problem 21
Internal
problem
ID
[8733]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
21
Date
solved
:
Wednesday, March 05, 2025 at 06:42:33 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _Chini]
\begin{align*} y^{\prime }&=\sqrt {y}+x \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 69
ode:=diff(y(x),x) = y(x)^(1/2)+x;
dsolve(ode,y(x), singsol=all);
\[
-\frac {2 \,\operatorname {arctanh}\left (2 \sqrt {\frac {y}{x^{2}}}\right )}{3}+\frac {4 \,\operatorname {arctanh}\left (\sqrt {\frac {y}{x^{2}}}\right )}{3}-\frac {\ln \left (\frac {-x^{2}+4 y}{x^{2}}\right )}{3}-\frac {2 \ln \left (2\right )}{3}-\frac {2 \ln \left (\frac {y-x^{2}}{x^{2}}\right )}{3}-2 \ln \left (x \right )+c_{1} = 0
\]
✓ Mathematica. Time used: 46.052 (sec). Leaf size: 716
ode=D[y[x],x]==Sqrt[y[x]]+x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{4} \left (3 x^2+\frac {e^{3 c_1} x \left (8+e^{3 c_1} x^3\right )}{\sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}}+e^{-6 c_1} \sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}\right ) \\
y(x)\to \frac {1}{72} \left (54 x^2-\frac {9 i \left (\sqrt {3}-i\right ) e^{3 c_1} x \left (8+e^{3 c_1} x^3\right )}{\sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}}+9 i \left (\sqrt {3}+i\right ) e^{-6 c_1} \sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}\right ) \\
y(x)\to \frac {1}{72} \left (54 x^2+\frac {9 i \left (\sqrt {3}+i\right ) e^{3 c_1} x \left (8+e^{3 c_1} x^3\right )}{\sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}}-9 \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{-e^{18 c_1} x^6+20 e^{15 c_1} x^3+8 \sqrt {-e^{24 c_1} \left (-1+e^{3 c_1} x^3\right ){}^3}+8 e^{12 c_1}}\right ) \\
y(x)\to \frac {-\left (-x^6\right )^{2/3}+3 x^4+\sqrt [3]{-x^6} x^2}{4 x^2} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (-x^6\right )^{2/3}+6 x^4+i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6} x^2}{8 x^2} \\
y(x)\to \frac {1}{8} x^2 \left (\frac {\left (1+i \sqrt {3}\right ) x^4}{\left (-x^6\right )^{2/3}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{-x^6}}+6\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x - sqrt(y(x)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -x - sqrt(y(x)) + Derivative(y(x), x) cannot be solved by the lie group method