60.1.414 problem 416

Internal problem ID [10428]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 416
Date solved : Monday, January 27, 2025 at 07:43:29 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}+\left (y-3 x \right ) y^{\prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 140

dsolve(x*diff(y(x),x)^2+(y(x)-3*x)*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
 

\begin{align*} y &= x \\ -\frac {c_{1} \left (5 x -y+\sqrt {9 x^{2}-10 x y+y^{2}}\right )}{x {\left (\frac {3 x -y+\sqrt {9 x^{2}-10 x y+y^{2}}}{x}\right )}^{{3}/{2}}}+x &= 0 \\ \frac {\left (-5 x +y+\sqrt {9 x^{2}-10 x y+y^{2}}\right ) c_{1} \sqrt {2}}{4 x {\left (\frac {-y+3 x -\sqrt {9 x^{2}-10 x y+y^{2}}}{x}\right )}^{{3}/{2}}}+x &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.130 (sec). Leaf size: 1221

DSolve[y[x] + (-3*x + y[x])*D[y[x],x] + x*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{12} \left (\frac {e^{8 c_1}-216 e^{4 c_1} x^2}{x^2 \sqrt [3]{-\frac {-5832 e^{4 c_1} x^4+540 e^{8 c_1} x^2-24 \sqrt {3} x \sqrt {e^{8 c_1} \left (27 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}}+\sqrt [3]{-\frac {-5832 e^{4 c_1} x^4+540 e^{8 c_1} x^2-24 \sqrt {3} x \sqrt {e^{8 c_1} \left (27 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}-\frac {e^{4 c_1}}{x}\right ) \\ y(x)\to \frac {1}{24} \left (\frac {\left (1+i \sqrt {3}\right ) \left (216 e^{4 c_1}-\frac {e^{8 c_1}}{x^2}\right )}{\sqrt [3]{-\frac {-5832 e^{4 c_1} x^4+540 e^{8 c_1} x^2-24 \sqrt {3} x \sqrt {e^{8 c_1} \left (27 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-\frac {-5832 e^{4 c_1} x^4+540 e^{8 c_1} x^2-24 \sqrt {3} x \sqrt {e^{8 c_1} \left (27 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}-\frac {2 e^{4 c_1}}{x}\right ) \\ y(x)\to \frac {1}{24} \left (\frac {\left (1-i \sqrt {3}\right ) \left (216 e^{4 c_1}-\frac {e^{8 c_1}}{x^2}\right )}{\sqrt [3]{-\frac {-5832 e^{4 c_1} x^4+540 e^{8 c_1} x^2-24 \sqrt {3} x \sqrt {e^{8 c_1} \left (27 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-\frac {-5832 e^{4 c_1} x^4+540 e^{8 c_1} x^2-24 \sqrt {3} x \sqrt {e^{8 c_1} \left (27 x^2+e^{4 c_1}\right ){}^3}+e^{12 c_1}}{x^3}}-\frac {2 e^{4 c_1}}{x}\right ) \\ y(x)\to \frac {1}{12} e^{-8 c_1} \left (\frac {e^{8 c_1}-216 e^{12 c_1} x^2}{x^2 \sqrt [3]{-\frac {-5832 e^{20 c_1} x^4+540 e^{16 c_1} x^2-24 \sqrt {3} x \sqrt {e^{28 c_1} \left (1+27 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}{x^3}}}+\sqrt [3]{-\frac {-5832 e^{20 c_1} x^4+540 e^{16 c_1} x^2-24 \sqrt {3} x \sqrt {e^{28 c_1} \left (1+27 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}{x^3}}-\frac {e^{4 c_1}}{x}\right ) \\ y(x)\to \frac {1}{24} e^{-8 c_1} \left (\frac {\left (1+i \sqrt {3}\right ) \left (216 e^{12 c_1}-\frac {e^{8 c_1}}{x^2}\right )}{\sqrt [3]{-\frac {-5832 e^{20 c_1} x^4+540 e^{16 c_1} x^2-24 \sqrt {3} x \sqrt {e^{28 c_1} \left (1+27 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}{x^3}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-\frac {-5832 e^{20 c_1} x^4+540 e^{16 c_1} x^2-24 \sqrt {3} x \sqrt {e^{28 c_1} \left (1+27 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}{x^3}}-\frac {2 e^{4 c_1}}{x}\right ) \\ y(x)\to \frac {1}{24} e^{-8 c_1} \left (\frac {\left (1-i \sqrt {3}\right ) \left (216 e^{12 c_1}-\frac {e^{8 c_1}}{x^2}\right )}{\sqrt [3]{-\frac {-5832 e^{20 c_1} x^4+540 e^{16 c_1} x^2-24 \sqrt {3} x \sqrt {e^{28 c_1} \left (1+27 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}{x^3}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-\frac {-5832 e^{20 c_1} x^4+540 e^{16 c_1} x^2-24 \sqrt {3} x \sqrt {e^{28 c_1} \left (1+27 e^{4 c_1} x^2\right ){}^3}+e^{12 c_1}}{x^3}}-\frac {2 e^{4 c_1}}{x}\right ) \\ \end{align*}