75.7.21 problem 196

Internal problem ID [16814]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 196
Date solved : Tuesday, January 28, 2025 at 09:33:36 AM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} 3 y^{2}-x +\left (2 y^{3}-6 y x \right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.485 (sec). Leaf size: 101

dsolve(( 3*y(x)^2-x)+( 2*y(x)^3-6*x*y(x) )*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= -\frac {\sqrt {-2 \sqrt {c_{1} \left (c_{1} -8 x \right )}+2 c_{1} -4 x}}{2} \\ y &= \frac {\sqrt {-2 \sqrt {c_{1} \left (c_{1} -8 x \right )}+2 c_{1} -4 x}}{2} \\ y &= -\frac {\sqrt {2 \sqrt {c_{1} \left (c_{1} -8 x \right )}+2 c_{1} -4 x}}{2} \\ y &= \frac {\sqrt {2 \sqrt {c_{1} \left (c_{1} -8 x \right )}+2 c_{1} -4 x}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.192 (sec). Leaf size: 112

DSolve[( 3*y[x]^2-x)+( 2*y[x]^3-6*x*y[x] )*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\left (\frac {2}{y(x)^2+K[1]}-\frac {1}{K[1]-y(x)^2}\right )dK[1]+\int _1^{y(x)}\left (-\frac {2 K[2]}{K[2]^2-x}+\frac {4 K[2]}{K[2]^2+x}-\int _1^x\left (-\frac {2 K[2]}{\left (K[1]-K[2]^2\right )^2}-\frac {4 K[2]}{\left (K[2]^2+K[1]\right )^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]