69.12.33 problem 307

Internal problem ID [18186]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 307
Date solved : Thursday, October 02, 2025 at 03:08:17 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 71

ode:=diff(y(x),x) = ((9*y(x)^2-6*y(x)+2)/(x^2-2*x+5))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ -\frac {\sqrt {\frac {9 y^{2}-6 y+2}{x^{2}-2 x +5}}\, \sqrt {x^{2}-2 x +5}\, \operatorname {arcsinh}\left (\frac {x}{2}-\frac {1}{2}\right )}{\sqrt {9 y^{2}-6 y+2}}+\frac {\operatorname {arcsinh}\left (3 y-1\right )}{3}+c_1 = 0 \]

✓ Mathematica. Time used: 6.97 (sec). Leaf size: 160

ode=D[y[x],x]==Sqrt[  (9*y[x]^2-6*y[x]+2)/ (x^2-2*x+5)   ]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{96} \left (e^{3 c_1} \left (x^3+\left (\sqrt {x^2-2 x+5}-3\right ) x^2-2 \left (\sqrt {x^2-2 x+5}-3\right ) x+2 \left (\sqrt {x^2-2 x+5}-2\right )\right )-64 e^{-3 c_1} \left (-x^3+\left (\sqrt {x^2-2 x+5}+3\right ) x^2-2 \left (\sqrt {x^2-2 x+5}+3\right ) x+2 \left (\sqrt {x^2-2 x+5}+2\right )\right )+32\right )\\ y(x)&\to \frac {1}{3}-\frac {i}{3}\\ y(x)&\to \frac {1}{3}+\frac {i}{3} \end{align*}

✓ Sympy. Time used: 2.179 (sec). Leaf size: 122

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt((9*y(x)**2 - 6*y(x) + 2)/(x**2 - 2*x + 5)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \begin {cases} \frac {y{\left (x \right )}}{\sqrt {9 y^{2}{\left (x \right )} - 6 y{\left (x \right )} + 2}} & \text {for}\: \left |{y{\left (x \right )}}\right | < 1 \\\frac {{G_{2, 2}^{1, 1}\left (\begin {matrix} 0 & 1 \\0 & -1 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{\sqrt {9 y^{2}{\left (x \right )} - 6 y{\left (x \right )} + 2}} + \frac {{G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{\sqrt {9 y^{2}{\left (x \right )} - 6 y{\left (x \right )} + 2}} & \text {otherwise} \end {cases} - \frac {\int \sqrt {\frac {9 y^{2}{\left (x \right )} - 6 y{\left (x \right )} + 2}{x^{2} - 2 x + 5}}\, dx}{\sqrt {9 y^{2}{\left (x \right )} - 6 y{\left (x \right )} + 2}} = C_{1} \]