40.5.15 problem 31

Internal problem ID [6680]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page 65
Problem number : 31
Date solved : Monday, January 27, 2025 at 02:20:54 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 2.283 (sec). Leaf size: 159

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

\begin{align*} y &= 0 \\ \ln \left (x \right )-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {-x^{2}+12 y^{2}}{x^{2}}}}\right )}{4}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-12 y^{2}}{x^{2}}}}{5}\right )}{4}+\frac {5 \ln \left (\frac {2 x^{2}+y^{2}}{x^{2}}\right )}{8}-\frac {\ln \left (\frac {y}{x}\right )}{4}-c_1 &= 0 \\ \ln \left (x \right )+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {-x^{2}+12 y^{2}}{x^{2}}}}\right )}{4}-\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-12 y^{2}}{x^{2}}}}{5}\right )}{4}+\frac {5 \ln \left (\frac {2 x^{2}+y^{2}}{x^{2}}\right )}{8}-\frac {\ln \left (\frac {y}{x}\right )}{4}-c_1 &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.269 (sec). Leaf size: 1131

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

\begin{align*} y(x)\to -\sqrt {\text {Root}\left [62208 \text {$\#$1}^5+622080 \text {$\#$1}^4 x^2+\text {$\#$1}^3 \left (2488320 x^4-864 e^{8 c_1}\right )+\text {$\#$1}^2 \left (4976640 x^6+16416 e^{8 c_1} x^2\right )+\text {$\#$1} \left (4976640 x^8-13968 e^{8 c_1} x^4+3 e^{16 c_1}\right )+1990656 x^{10}-512 e^{8 c_1} x^6\&,1\right ]} \\ y(x)\to \sqrt {\text {Root}\left [62208 \text {$\#$1}^5+622080 \text {$\#$1}^4 x^2+\text {$\#$1}^3 \left (2488320 x^4-864 e^{8 c_1}\right )+\text {$\#$1}^2 \left (4976640 x^6+16416 e^{8 c_1} x^2\right )+\text {$\#$1} \left (4976640 x^8-13968 e^{8 c_1} x^4+3 e^{16 c_1}\right )+1990656 x^{10}-512 e^{8 c_1} x^6\&,1\right ]} \\ y(x)\to -\sqrt {\text {Root}\left [62208 \text {$\#$1}^5+622080 \text {$\#$1}^4 x^2+\text {$\#$1}^3 \left (2488320 x^4-864 e^{8 c_1}\right )+\text {$\#$1}^2 \left (4976640 x^6+16416 e^{8 c_1} x^2\right )+\text {$\#$1} \left (4976640 x^8-13968 e^{8 c_1} x^4+3 e^{16 c_1}\right )+1990656 x^{10}-512 e^{8 c_1} x^6\&,2\right ]} \\ y(x)\to \sqrt {\text {Root}\left [62208 \text {$\#$1}^5+622080 \text {$\#$1}^4 x^2+\text {$\#$1}^3 \left (2488320 x^4-864 e^{8 c_1}\right )+\text {$\#$1}^2 \left (4976640 x^6+16416 e^{8 c_1} x^2\right )+\text {$\#$1} \left (4976640 x^8-13968 e^{8 c_1} x^4+3 e^{16 c_1}\right )+1990656 x^{10}-512 e^{8 c_1} x^6\&,2\right ]} \\ y(x)\to -\sqrt {\text {Root}\left [62208 \text {$\#$1}^5+622080 \text {$\#$1}^4 x^2+\text {$\#$1}^3 \left (2488320 x^4-864 e^{8 c_1}\right )+\text {$\#$1}^2 \left (4976640 x^6+16416 e^{8 c_1} x^2\right )+\text {$\#$1} \left (4976640 x^8-13968 e^{8 c_1} x^4+3 e^{16 c_1}\right )+1990656 x^{10}-512 e^{8 c_1} x^6\&,3\right ]} \\ y(x)\to \sqrt {\text {Root}\left [62208 \text {$\#$1}^5+622080 \text {$\#$1}^4 x^2+\text {$\#$1}^3 \left (2488320 x^4-864 e^{8 c_1}\right )+\text {$\#$1}^2 \left (4976640 x^6+16416 e^{8 c_1} x^2\right )+\text {$\#$1} \left (4976640 x^8-13968 e^{8 c_1} x^4+3 e^{16 c_1}\right )+1990656 x^{10}-512 e^{8 c_1} x^6\&,3\right ]} \\ y(x)\to -\sqrt {\text {Root}\left [62208 \text {$\#$1}^5+622080 \text {$\#$1}^4 x^2+\text {$\#$1}^3 \left (2488320 x^4-864 e^{8 c_1}\right )+\text {$\#$1}^2 \left (4976640 x^6+16416 e^{8 c_1} x^2\right )+\text {$\#$1} \left (4976640 x^8-13968 e^{8 c_1} x^4+3 e^{16 c_1}\right )+1990656 x^{10}-512 e^{8 c_1} x^6\&,4\right ]} \\ y(x)\to \sqrt {\text {Root}\left [62208 \text {$\#$1}^5+622080 \text {$\#$1}^4 x^2+\text {$\#$1}^3 \left (2488320 x^4-864 e^{8 c_1}\right )+\text {$\#$1}^2 \left (4976640 x^6+16416 e^{8 c_1} x^2\right )+\text {$\#$1} \left (4976640 x^8-13968 e^{8 c_1} x^4+3 e^{16 c_1}\right )+1990656 x^{10}-512 e^{8 c_1} x^6\&,4\right ]} \\ y(x)\to -\sqrt {\text {Root}\left [62208 \text {$\#$1}^5+622080 \text {$\#$1}^4 x^2+\text {$\#$1}^3 \left (2488320 x^4-864 e^{8 c_1}\right )+\text {$\#$1}^2 \left (4976640 x^6+16416 e^{8 c_1} x^2\right )+\text {$\#$1} \left (4976640 x^8-13968 e^{8 c_1} x^4+3 e^{16 c_1}\right )+1990656 x^{10}-512 e^{8 c_1} x^6\&,5\right ]} \\ y(x)\to \sqrt {\text {Root}\left [62208 \text {$\#$1}^5+622080 \text {$\#$1}^4 x^2+\text {$\#$1}^3 \left (2488320 x^4-864 e^{8 c_1}\right )+\text {$\#$1}^2 \left (4976640 x^6+16416 e^{8 c_1} x^2\right )+\text {$\#$1} \left (4976640 x^8-13968 e^{8 c_1} x^4+3 e^{16 c_1}\right )+1990656 x^{10}-512 e^{8 c_1} x^6\&,5\right ]} \\ \end{align*}