60.1.382 problem 391

Internal problem ID [10396]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 391
Date solved : Wednesday, March 05, 2025 at 10:44:13 AM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}+\left (a y+b x \right ) y^{\prime }+a b x y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 22

ode:=diff(y(x),x)^2+(b*x+a*y(x))*diff(y(x),x)+a*b*x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_{1} {\mathrm e}^{-a x} \\ y &= -\frac {b \,x^{2}}{2}+c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 34

ode=a*b*x*y[x] + (b*x + a*y[x])*D[y[x],x] + D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1 e^{-a x} \\ y(x)\to -\frac {b x^2}{2}+c_1 \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.186 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*b*x*y(x) + (a*y(x) + b*x)*Derivative(y(x), x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} e^{- a x}, \ y{\left (x \right )} = C_{1} - \frac {b x^{2}}{2}\right ] \]