62.14.1 problem Ex 1

Internal problem ID [12882]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 25. Equations solvable for \(y\). Page 52
Problem number : Ex 1
Date solved : Tuesday, January 28, 2025 at 04:31:43 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 69

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

\begin{align*} y &= -1+\sqrt {4 c_{1} x +1}-\ln \left (2\right )+\ln \left (\frac {-1+\sqrt {4 c_{1} x +1}}{x}\right ) \\ y &= -1-\sqrt {4 c_{1} x +1}-\ln \left (2\right )+\ln \left (\frac {-1-\sqrt {4 c_{1} x +1}}{x}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.143 (sec). Leaf size: 258

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

\[ \text {Solve}\left [\int _1^x\frac {W\left (2 e^{y(x)} K[1]\right )}{K[1] \left (W\left (2 e^{y(x)} K[1]\right )+2\right )}dK[1]+\int _1^{y(x)}-\frac {W\left (2 e^{K[2]} x\right ) \int _1^x\left (\frac {W\left (2 e^{K[2]} K[1]\right )}{K[1] \left (W\left (2 e^{K[2]} K[1]\right )+1\right ) \left (W\left (2 e^{K[2]} K[1]\right )+2\right )}-\frac {W\left (2 e^{K[2]} K[1]\right )^2}{K[1] \left (W\left (2 e^{K[2]} K[1]\right )+1\right ) \left (W\left (2 e^{K[2]} K[1]\right )+2\right )^2}\right )dK[1]+2 \int _1^x\left (\frac {W\left (2 e^{K[2]} K[1]\right )}{K[1] \left (W\left (2 e^{K[2]} K[1]\right )+1\right ) \left (W\left (2 e^{K[2]} K[1]\right )+2\right )}-\frac {W\left (2 e^{K[2]} K[1]\right )^2}{K[1] \left (W\left (2 e^{K[2]} K[1]\right )+1\right ) \left (W\left (2 e^{K[2]} K[1]\right )+2\right )^2}\right )dK[1]+2}{W\left (2 e^{K[2]} x\right )+2}dK[2]=c_1,y(x)\right ] \]