60.1.418 problem 420

Internal problem ID [10432]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 420
Date solved : Monday, January 27, 2025 at 07:43:41 PM
CAS classification : [[_homogeneous, `class G`], _rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}-2 y^{\prime } y+a&=0 \end{align*}

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 788

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

\begin{align*} y &= \frac {\left (\frac {4 x^{2}}{\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}}+2 x +\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}\right ) x}{12 c_{1}}+\frac {3 a c_{1} \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}}{\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}+2 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}+4 x^{2}} \\ y &= -\frac {\left (\left (1+i \sqrt {3}\right ) \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}-4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}-4 \left (i \sqrt {3}-1\right ) x^{2}\right ) x}{24 \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}} c_{1}}+\frac {6 a c_{1} \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}}{4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}-4 x^{2}+4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}-\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}} \\ y &= \frac {\left (\left (i \sqrt {3}-1\right ) \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}+4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}-4 x^{2} \left (1+i \sqrt {3}\right )\right ) x}{24 \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}} c_{1}}-\frac {6 a c_{1} \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}}{-i \sqrt {3}\, \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}+4 i \sqrt {3}\, x^{2}+\left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{2}/{3}}-4 x \left (-36 a \,c_{1}^{2}+8 x^{3}+12 \sqrt {a \left (9 a \,c_{1}^{2}-4 x^{3}\right )}\, c_{1} \right )^{{1}/{3}}+4 x^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.123 (sec). Leaf size: 1553

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

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