29.23.1 problem 631

Internal problem ID [5223]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 23
Problem number : 631
Date solved : Monday, January 27, 2025 at 10:23:46 AM
CAS classification : [_exact, _rational]

\begin{align*} \left (1-3 x^{2} y+6 y^{2}\right ) y^{\prime }+x^{2}-3 x y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 769

dsolve((1-3*x^2*y(x)+6*y(x)^2)*diff(y(x),x)+x^2-3*x*y(x)^2 = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 c_{1} x^{3}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}}{12}+\frac {3 x^{4}-8}{4 \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 c_{1} x^{3}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}}+\frac {x^{2}}{4} \\ y \left (x \right ) &= \frac {24+i \left (-24+9 x^{4}-\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{2}/{3}}\right ) \sqrt {3}-9 x^{4}+6 x^{2} \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}-\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{2}/{3}}}{24 \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {24+i \left (-9 x^{4}+\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{2}/{3}}+24\right ) \sqrt {3}-9 x^{4}+6 x^{2} \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}-\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{2}/{3}}}{24 \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 7.965 (sec). Leaf size: 570

DSolve[(1-3 x^2 y[x]+6 y[x]^2)D[y[x],x]+x^2-3 x y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {x^2}{4}-\frac {\sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}{6 \sqrt [3]{2}}+\frac {6-\frac {9 x^4}{4}}{3\ 2^{2/3} \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}} \\ y(x)\to \frac {x^2}{4}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}-\frac {\left (1+i \sqrt {3}\right ) \left (6-\frac {9 x^4}{4}\right )}{6\ 2^{2/3} \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}} \\ y(x)\to \frac {x^2}{4}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}-\frac {\left (1-i \sqrt {3}\right ) \left (6-\frac {9 x^4}{4}\right )}{6\ 2^{2/3} \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}} \\ \end{align*}