60.1.550 problem 563

Internal problem ID [10564]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 563
Date solved : Sunday, March 30, 2025 at 06:07:27 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*}

✓ Maple. Time used: 0.179 (sec). Leaf size: 54

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

\[ -{\left (\frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-a y-b}\right )}{x}\right )}^{-\frac {1}{a +1}} c_1 +x -\frac {x}{a \operatorname {LambertW}\left (x \,{\mathrm e}^{-a y-b}\right )} = 0 \]

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

ode=b + Log[D[y[x],x]] + a*y[x] + x*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 2.859 (sec). Leaf size: 97

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*y(x) + b + x*Derivative(y(x), x) + log(Derivative(y(x), x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - a \left (\begin {cases} - y{\left (x \right )} + \frac {\log {\left (x \right )}}{a} - \frac {\log {\left (W\left (e^{- b} e^{- a \left (y{\left (x \right )} - \frac {\log {\left (x \right )}}{a}\right )}\right ) - \frac {1}{a} \right )}}{a} - \frac {W\left (e^{- b} e^{- a \left (y{\left (x \right )} - \frac {\log {\left (x \right )}}{a}\right )}\right )}{a} - \frac {\log {\left (W\left (e^{- b} e^{- a \left (y{\left (x \right )} - \frac {\log {\left (x \right )}}{a}\right )}\right ) - \frac {1}{a} \right )}}{a^{2}} & \text {for}\: a \neq 0 \\- y{\left (x \right )} + \frac {\log {\left (x \right )}}{a} & \text {otherwise} \end {cases}\right ) + \log {\left (x \right )} = 0 \]