15.20.10 problem 10

Internal problem ID [3318]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 38, page 173
Problem number : 10
Date solved : Monday, January 27, 2025 at 07:33:24 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y&=y^{\prime } x \left (y^{\prime }+1\right ) \end{align*}

✓ Solution by Maple

Time used: 0.030 (sec). Leaf size: 65

dsolve(y(x)=diff(y(x),x)*x*(diff(y(x),x)+1),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.523 (sec). Leaf size: 102

DSolve[y[x]==D[y[x],x]*x*(D[y[x],x]+1),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\frac {1}{\sqrt {\frac {4 y(x)}{x}+1}-1}-\log \left (\sqrt {\frac {4 y(x)}{x}+1}-1\right )&=\frac {\log (x)}{2}+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {1}{\sqrt {\frac {4 y(x)}{x}+1}+1}+\log \left (\sqrt {\frac {4 y(x)}{x}+1}+1\right )&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}