29.30.5 problem 863

Internal problem ID [5445]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 30
Problem number : 863
Date solved : Monday, January 27, 2025 at 11:24:05 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 139

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

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

✓ Solution by Mathematica

Time used: 60.738 (sec). Leaf size: 1221

DSolve[x (D[y[x],x])^2-(3 x-y[x])D[y[x],x]+y[x]==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*}