11.10 problem 269

Internal problem ID [15131]

Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 11. Singular solutions of differential equations. Exercises page 92
Problem number: 269.
ODE order: 1.
ODE degree: 2.

CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]

\[ \boxed {y-{y^{\prime }}^{2}+x y^{\prime }=x} \]

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 845

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

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

✓ Solution by Mathematica

Time used: 61.116 (sec). Leaf size: 2409

DSolve[y[x]==y'[x]^2-x*y'[x]+x^2/x,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {2 \sqrt [3]{2} x^4-8 \sqrt [3]{2} x^3+12 \sqrt [3]{2} x^2+4 x^2 \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3-10 e^{3 c_1} x^3-30 x^2+30 e^{3 c_1} x^2+12 x-30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (4 (x-1)^3+e^{3 c_1}\right ){}^3}-2+10 e^{3 c_1}+e^{6 c_1}}-4 x \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3-10 e^{3 c_1} x^3-30 x^2+30 e^{3 c_1} x^2+12 x-30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (4 (x-1)^3+e^{3 c_1}\right ){}^3}-2+10 e^{3 c_1}+e^{6 c_1}}+2^{2/3} \left (-2 x^6+12 x^5-30 x^4+40 x^3-10 e^{3 c_1} x^3-30 x^2+30 e^{3 c_1} x^2+12 x-30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (4 (x-1)^3+e^{3 c_1}\right ){}^3}-2+10 e^{3 c_1}+e^{6 c_1}\right ){}^{2/3}+6 \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3-10 e^{3 c_1} x^3-30 x^2+30 e^{3 c_1} x^2+12 x-30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (4 (x-1)^3+e^{3 c_1}\right ){}^3}-2+10 e^{3 c_1}+e^{6 c_1}}-8 \sqrt [3]{2} x-4 \sqrt [3]{2} e^{3 c_1} x+2 \sqrt [3]{2}+4 \sqrt [3]{2} e^{3 c_1}}{8 \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3-10 e^{3 c_1} x^3-30 x^2+30 e^{3 c_1} x^2+12 x-30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (4 (x-1)^3+e^{3 c_1}\right ){}^3}-2+10 e^{3 c_1}+e^{6 c_1}}} \\ y(x)\to \frac {1}{4} \left (2 x^2-2 x+3\right )+\frac {\left (1+i \sqrt {3}\right ) (x-1) \left (-(x-1)^3+2 e^{3 c_1}\right )}{4\ 2^{2/3} \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3-10 e^{3 c_1} x^3-30 x^2+30 e^{3 c_1} x^2+12 x-30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (4 (x-1)^3+e^{3 c_1}\right ){}^3}-2+10 e^{3 c_1}+e^{6 c_1}}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3-10 e^{3 c_1} x^3-30 x^2+30 e^{3 c_1} x^2+12 x-30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (4 (x-1)^3+e^{3 c_1}\right ){}^3}-2+10 e^{3 c_1}+e^{6 c_1}}}{8 \sqrt [3]{2}} \\ y(x)\to \frac {1}{4} \left (2 x^2-2 x+3\right )+\frac {i \left (\sqrt {3}+i\right ) (x-1) \left ((x-1)^3-2 e^{3 c_1}\right )}{4\ 2^{2/3} \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3-10 e^{3 c_1} x^3-30 x^2+30 e^{3 c_1} x^2+12 x-30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (4 (x-1)^3+e^{3 c_1}\right ){}^3}-2+10 e^{3 c_1}+e^{6 c_1}}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3-10 e^{3 c_1} x^3-30 x^2+30 e^{3 c_1} x^2+12 x-30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (4 (x-1)^3+e^{3 c_1}\right ){}^3}-2+10 e^{3 c_1}+e^{6 c_1}}}{8 \sqrt [3]{2}} \\ y(x)\to \frac {2 \sqrt [3]{2} x^4-8 \sqrt [3]{2} x^3+12 \sqrt [3]{2} x^2+4 x^2 \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3+10 e^{3 c_1} x^3-30 x^2-30 e^{3 c_1} x^2+12 x+30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (-4 (x-1)^3+e^{3 c_1}\right ){}^3}-2-10 e^{3 c_1}+e^{6 c_1}}-4 x \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3+10 e^{3 c_1} x^3-30 x^2-30 e^{3 c_1} x^2+12 x+30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (-4 (x-1)^3+e^{3 c_1}\right ){}^3}-2-10 e^{3 c_1}+e^{6 c_1}}+2^{2/3} \left (-2 x^6+12 x^5-30 x^4+40 x^3+10 e^{3 c_1} x^3-30 x^2-30 e^{3 c_1} x^2+12 x+30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (-4 (x-1)^3+e^{3 c_1}\right ){}^3}-2-10 e^{3 c_1}+e^{6 c_1}\right ){}^{2/3}+6 \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3+10 e^{3 c_1} x^3-30 x^2-30 e^{3 c_1} x^2+12 x+30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (-4 (x-1)^3+e^{3 c_1}\right ){}^3}-2-10 e^{3 c_1}+e^{6 c_1}}-8 \sqrt [3]{2} x+4 \sqrt [3]{2} e^{3 c_1} x+2 \sqrt [3]{2}-4 \sqrt [3]{2} e^{3 c_1}}{8 \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3+10 e^{3 c_1} x^3-30 x^2-30 e^{3 c_1} x^2+12 x+30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (-4 (x-1)^3+e^{3 c_1}\right ){}^3}-2-10 e^{3 c_1}+e^{6 c_1}}} \\ y(x)\to \frac {1}{4} \left (2 x^2-2 x+3\right )-\frac {i \left (\sqrt {3}-i\right ) (x-1) \left ((x-1)^3+2 e^{3 c_1}\right )}{4\ 2^{2/3} \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3+10 e^{3 c_1} x^3-30 x^2-30 e^{3 c_1} x^2+12 x+30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (-4 (x-1)^3+e^{3 c_1}\right ){}^3}-2-10 e^{3 c_1}+e^{6 c_1}}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3+10 e^{3 c_1} x^3-30 x^2-30 e^{3 c_1} x^2+12 x+30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (-4 (x-1)^3+e^{3 c_1}\right ){}^3}-2-10 e^{3 c_1}+e^{6 c_1}}}{8 \sqrt [3]{2}} \\ y(x)\to \frac {1}{4} \left (2 x^2-2 x+3\right )+\frac {i \left (\sqrt {3}+i\right ) (x-1) \left ((x-1)^3+2 e^{3 c_1}\right )}{4\ 2^{2/3} \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3+10 e^{3 c_1} x^3-30 x^2-30 e^{3 c_1} x^2+12 x+30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (-4 (x-1)^3+e^{3 c_1}\right ){}^3}-2-10 e^{3 c_1}+e^{6 c_1}}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{-2 x^6+12 x^5-30 x^4+40 x^3+10 e^{3 c_1} x^3-30 x^2-30 e^{3 c_1} x^2+12 x+30 e^{3 c_1} x+\sqrt {e^{3 c_1} \left (-4 (x-1)^3+e^{3 c_1}\right ){}^3}-2-10 e^{3 c_1}+e^{6 c_1}}}{8 \sqrt [3]{2}} \\ \end{align*}