77.1.64 problem 83 (page 120)

Internal problem ID [17954]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 83 (page 120)
Date solved : Tuesday, January 28, 2025 at 11:18:14 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Solution by Maple

Time used: 0.153 (sec). Leaf size: 77

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

\begin{align*} -\frac {c_{1}}{\sqrt {-x -\sqrt {x^{2}-2 y}}}+\frac {2 x}{3}-\frac {\sqrt {x^{2}-2 y}}{3} &= 0 \\ -\frac {c_{1}}{\sqrt {-x +\sqrt {x^{2}-2 y}}}+\frac {2 x}{3}+\frac {\sqrt {x^{2}-2 y}}{3} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.167 (sec). Leaf size: 1000

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

\begin{align*} y(x)\to \frac {\left (x^2-\sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}\right ){}^2-4 e^{3 c_1} x}{2 \sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}} \\ y(x)\to \frac {1}{4} \left (-4 x^2+\frac {\left (1+i \sqrt {3}\right ) x \left (-x^3+4 e^{3 c_1}\right )}{\sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}\right ) \\ y(x)\to -x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3-4 e^{3 c_1}\right )}{4 \sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}}}-\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [3]{x^6+10 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (2 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}} \\ y(x)\to \frac {2 \sqrt [3]{2} x^4+\left (4 x^6+20 e^{3 c_1} x^3+2 \sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-2 e^{6 c_1}\right ){}^{2/3}-4 x^2 \sqrt [3]{2 x^6+10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-e^{6 c_1}}-4 \sqrt [3]{2} e^{3 c_1} x}{4 \sqrt [3]{2 x^6+10 e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-e^{6 c_1}}} \\ y(x)\to -x^2+\frac {\left (1+i \sqrt {3}\right ) x \left (-x^3+2 e^{3 c_1}\right )}{4 \sqrt [3]{x^6+5 e^{3 c_1} x^3+\frac {1}{2} \sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-\frac {e^{6 c_1}}{2}}}+\frac {1}{4} i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+5 e^{3 c_1} x^3+\frac {1}{2} \sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-\frac {e^{6 c_1}}{2}} \\ y(x)\to -x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3-2 e^{3 c_1}\right )}{4 \sqrt [3]{x^6+5 e^{3 c_1} x^3+\frac {1}{2} \sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-\frac {e^{6 c_1}}{2}}}-\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [3]{x^6+5 e^{3 c_1} x^3+\frac {1}{2} \sqrt {e^{3 c_1} \left (4 x^3+e^{3 c_1}\right ){}^3}-\frac {e^{6 c_1}}{2}} \\ \end{align*}