75.12.33 problem 307

Internal problem ID [16899]
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 : Tuesday, January 28, 2025 at 09:40:12 AM
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*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 71

dsolve(diff(y(x),x)=sqrt(  (9*y(x)^2-6*y(x)+2)/ (x^2-2*x+5)   ),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 5.760 (sec). Leaf size: 160

DSolve[D[y[x],x]==Sqrt[  (9*y[x]^2-6*y[x]+2)/ (x^2-2*x+5)   ],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*}