29.37.23 problem 1145

Internal problem ID [5681]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1145
Date solved : Monday, January 27, 2025 at 01:06:40 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Solution by Maple

Time used: 0.273 (sec). Leaf size: 73

dsolve(ln(diff(y(x),x))+x*diff(y(x),x)+a+b*y(x) = 0,y(x), singsol=all)
 

\[ \frac {-\left ({\left (\frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-b y \left (x \right )-a}\right )}{x}\right )}^{-\frac {1}{b +1}} c_{1} -x \right ) b \operatorname {LambertW}\left (x \,{\mathrm e}^{-b y \left (x \right )-a}\right )-x}{b \operatorname {LambertW}\left (x \,{\mathrm e}^{-b y \left (x \right )-a}\right )} = 0 \]

✓ Solution by Mathematica

Time used: 0.139 (sec). Leaf size: 59

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

\[ \text {Solve}\left [b \left (\frac {(b+1) \log \left (1-b W\left (x e^{-a-b y(x)}\right )\right )}{b^2}+\frac {W\left (x e^{-a-b y(x)}\right )}{b}\right )+b y(x)=c_1,y(x)\right ] \]