75.7.7 problem 181

Internal problem ID [16800]
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 : 181
Date solved : Tuesday, January 28, 2025 at 09:29:28 AM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 635

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

\begin{align*} y &= \frac {\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+162 c_{1} x^{3}+135 x^{4}-162 c_{1} x^{2}-54 x^{3}+81 c_{1}^{2}+54 c_{1} x -15 x^{2}+12 c_{1}}\right )^{{1}/{3}}}{6}+\frac {2 x +\frac {2}{3}}{\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+162 c_{1} x^{3}+135 x^{4}-162 c_{1} x^{2}-54 x^{3}+81 c_{1}^{2}+54 c_{1} x -15 x^{2}+12 c_{1}}\right )^{{1}/{3}}}-\frac {1}{3} \\ y &= \frac {i \left (4-\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{{2}/{3}}+12 x \right ) \sqrt {3}-12 x -{\left (\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}+2\right )}^{2}}{12 \left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}} \\ y &= \frac {i \left (\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{{2}/{3}}-12 x -4\right ) \sqrt {3}-12 x -{\left (\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}+2\right )}^{2}}{12 \left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.835 (sec). Leaf size: 478

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

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