62.38.3 problem Ex 3

Internal problem ID [13014]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 62. Summary. Page 144
Problem number : Ex 3
Date solved : Tuesday, January 28, 2025 at 04:49:16 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.221 (sec). Leaf size: 23

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

\[ y = \frac {\tanh \left (\frac {\left (x +c_{2} \right ) \sqrt {2}}{2 c_{1}}\right ) \sqrt {2}}{c_{1}} \]

✓ Solution by Mathematica

Time used: 0.137 (sec). Leaf size: 96

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{c_1-\frac {K[1]^2}{2}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {1}{2} K[1]^2-c_1}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{c_1-\frac {K[1]^2}{2}}dK[1]\&\right ][x+c_2] \\ \end{align*}