60.1.560 problem 563

Internal problem ID [10574]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 563
Date solved : Monday, January 27, 2025 at 09:16:17 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

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

✓ Solution by Maple

Time used: 0.266 (sec). Leaf size: 73

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

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

✓ Solution by Mathematica

Time used: 0.193 (sec). Leaf size: 380

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

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {W\left (e^{-b-a K[2]} x\right ) \int _1^x\left (\frac {a^3 W\left (e^{-b-a K[2]} K[1]\right )^2}{K[1] \left (W\left (e^{-b-a K[2]} K[1]\right )+1\right ) \left (a W\left (e^{-b-a K[2]} K[1]\right )-1\right )^2}-\frac {a^2 W\left (e^{-b-a K[2]} K[1]\right )}{K[1] \left (W\left (e^{-b-a K[2]} K[1]\right )+1\right ) \left (a W\left (e^{-b-a K[2]} K[1]\right )-1\right )}\right )dK[1] a+a-\int _1^x\left (\frac {a^3 W\left (e^{-b-a K[2]} K[1]\right )^2}{K[1] \left (W\left (e^{-b-a K[2]} K[1]\right )+1\right ) \left (a W\left (e^{-b-a K[2]} K[1]\right )-1\right )^2}-\frac {a^2 W\left (e^{-b-a K[2]} K[1]\right )}{K[1] \left (W\left (e^{-b-a K[2]} K[1]\right )+1\right ) \left (a W\left (e^{-b-a K[2]} K[1]\right )-1\right )}\right )dK[1]}{a W\left (e^{-b-a K[2]} x\right )-1}dK[2]+\int _1^x\frac {a W\left (e^{-b-a y(x)} K[1]\right )}{K[1] \left (a W\left (e^{-b-a y(x)} K[1]\right )-1\right )}dK[1]=c_1,y(x)\right ] \]