Internal problem ID [4094]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 855.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {{y^{\prime }}^{2} x -y y^{\prime }=-a x} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 50
dsolve(x*diff(y(x),x)^2-y(x)*diff(y(x),x)+a*x = 0,y(x), singsol=all)
\[ y \left (x \right ) = \left (-\operatorname {LambertW}\left (-\frac {x^{2}}{c_{1}^{2} a}\right )+1\right ) a c_{1} \sqrt {-\frac {x^{2}}{c_{1}^{2} a \operatorname {LambertW}\left (-\frac {x^{2}}{c_{1}^{2} a}\right )}} \]
✓ Solution by Mathematica
Time used: 0.924 (sec). Leaf size: 167
DSolve[x (y'[x])^2-y[x] y'[x]+a x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {4 a \arctan \left (\frac {y(x)}{x \sqrt {4 a-\frac {y(x)^2}{x^2}}}\right )+\frac {y(x) \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}-\frac {i y(x)}{x}\right )}{x}}{8 a}&=\frac {1}{2} i \log (x)+c_1,y(x)\right ] \\ \text {Solve}\left [\frac {4 a \arctan \left (\frac {y(x)}{x \sqrt {4 a-\frac {y(x)^2}{x^2}}}\right )+\frac {y(x) \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}+\frac {i y(x)}{x}\right )}{x}}{8 a}&=c_1-\frac {1}{2} i \log (x),y(x)\right ] \\ \end{align*}