77.1.73 problem 92 (page 135)

Internal problem ID [17963]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 92 (page 135)
Date solved : Tuesday, January 28, 2025 at 11:18:37 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 31

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

\begin{align*} y &= -x \\ y &= c_{1} +2 \sqrt {c_{1} x} \\ y &= c_{1} -2 \sqrt {c_{1} x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.096 (sec). Leaf size: 1178

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

\begin{align*} y(x)\to \frac {\sqrt [3]{2} \sqrt {x} \left (x^{3/2}-2 e^{\frac {3 c_1}{2}}\right )}{\sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}+\frac {\sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}{\sqrt [3]{2}}+2 x \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt {x} \left (-x^{3/2}+2 e^{\frac {3 c_1}{2}}\right )}{2^{2/3} \sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}{2 \sqrt [3]{2}}+2 x \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt {x} \left (x^{3/2}-2 e^{\frac {3 c_1}{2}}\right )}{2^{2/3} \sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-10 e^{\frac {3 c_1}{2}} x^{3/2}+\sqrt {e^{\frac {3 c_1}{2}} \left (4 x^{3/2}+e^{\frac {3 c_1}{2}}\right ){}^3}-2 x^3+e^{3 c_1}}}{2 \sqrt [3]{2}}+2 x \\ y(x)\to \frac {\sqrt [3]{2} e^{\frac {3 c_1}{2}} \sqrt {x} \left (2+e^{\frac {3 c_1}{2}} x^{3/2}\right )}{\sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}}+\frac {e^{-3 c_1} \sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}}{\sqrt [3]{2}}+2 x \\ y(x)\to \frac {1}{4} \left (-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) e^{\frac {3 c_1}{2}} \sqrt {x} \left (2+e^{\frac {3 c_1}{2}} x^{3/2}\right )}{\sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}}+i 2^{2/3} \left (\sqrt {3}+i\right ) e^{-3 c_1} \sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}+8 x\right ) \\ y(x)\to \frac {1}{4} \left (\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) e^{\frac {3 c_1}{2}} \sqrt {x} \left (2+e^{\frac {3 c_1}{2}} x^{3/2}\right )}{\sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}}-2^{2/3} \left (1+i \sqrt {3}\right ) e^{-3 c_1} \sqrt [3]{10 e^{\frac {15 c_1}{2}} x^{3/2}+\sqrt {-e^{12 c_1} \left (-1+4 e^{\frac {3 c_1}{2}} x^{3/2}\right ){}^3}-2 e^{9 c_1} x^3+e^{6 c_1}}+8 x\right ) \\ \end{align*}