1.62 problem 89

Internal problem ID [12159]

Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 89.
ODE order: 1.
ODE degree: 2.

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

\[ \boxed {y-2 x y^{\prime }-{y^{\prime }}^{2}=0} \]

Solution by Maple

Time used: 0.016 (sec). Leaf size: 690

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

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

Solution by Mathematica

Time used: 60.162 (sec). Leaf size: 931

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

\begin{align*} y(x)\to \frac {1}{4} \left (-x^2+\frac {x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}+\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ) y(x)\to \frac {1}{72} \left (-18 x^2-\frac {9 i \left (\sqrt {3}-i\right ) x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}+9 i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ) y(x)\to \frac {1}{72} \left (-18 x^2+\frac {9 i \left (\sqrt {3}+i\right ) x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}-9 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ) y(x)\to \frac {1}{4} \left (-x^2+\frac {x \left (x^3-8 e^{3 c_1}\right )}{\sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}+\sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ) y(x)\to \frac {1}{72} \left (-18 x^2+\frac {9 \left (1+i \sqrt {3}\right ) x \left (-x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}+9 i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ) y(x)\to \frac {1}{72} \left (-18 x^2+\frac {9 i \left (\sqrt {3}+i\right ) x \left (x^3-8 e^{3 c_1}\right )}{\sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}-9 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ) \end{align*}