83.45.15 problem Ex 15 page 59

Internal problem ID [19479]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter IV. Equations of the first order but not of the first degree
Problem number : Ex 15 page 59
Date solved : Thursday, March 13, 2025 at 02:35:27 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y-2 x y^{\prime }+a y {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 94

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

\begin{align*} y \left (x \right ) &= \frac {x}{\sqrt {a}} \\ y \left (x \right ) &= -\frac {x}{\sqrt {a}} \\ y \left (x \right ) &= 0 \\ \ln \left (x \right )+\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {a y \left (x \right )^{2}-x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-c_{1} &= 0 \\ \ln \left (x \right )-\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {a y \left (x \right )^{2}-x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-c_{1} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 2.053 (sec). Leaf size: 168

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

\begin{align*} y(x)\to -\frac {e^{\frac {a c_1}{2}} \sqrt {e^{a c_1}+2 i x}}{\sqrt {a}} \\ y(x)\to \frac {e^{\frac {a c_1}{2}} \sqrt {e^{a c_1}+2 i x}}{\sqrt {a}} \\ y(x)\to -\frac {e^{\frac {a c_1}{2}} \sqrt {e^{a c_1}-2 i a x}}{a^{3/2}} \\ y(x)\to \frac {e^{\frac {a c_1}{2}} \sqrt {e^{a c_1}-2 i a x}}{a^{3/2}} \\ y(x)\to -\frac {x}{\sqrt {a}} \\ y(x)\to \frac {x}{\sqrt {a}} \\ \end{align*}

✗ Sympy

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

Timed Out