75.7.13 problem 188

Internal problem ID [16806]
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 : 188
Date solved : Tuesday, January 28, 2025 at 09:31:30 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.122 (sec). Leaf size: 270

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

\begin{align*} y &= -\frac {12^{{1}/{3}} \left (x^{4} c_{1}^{2} 12^{{1}/{3}}-{\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{{2}/{3}}\right )}{6 c_{1} x {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{{1}/{3}}} \\ y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{{2}/{3}}+2^{{2}/{3}} x^{4} c_{1}^{2} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right )}{12 {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{{1}/{3}} x c_{1}} \\ y &= \frac {2^{{2}/{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{{2}/{3}}+2^{{2}/{3}} x^{4} c_{1}^{2} \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right )\right ) 3^{{1}/{3}}}{12 {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{{1}/{3}} x c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.118 (sec). Leaf size: 44

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {3 K[1]^2+1}{K[1] \left (K[1]^2+1\right )}dK[1]=-4 \log (x)+c_1,y(x)\right ] \]