75.7.8 problem 182

Internal problem ID [16730]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 182
Date solved : Monday, March 31, 2025 at 03:12:55 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 64

ode:=x*y(x)/(x^2+1)^(1/2)+2*x*y(x)-y(x)/x+((x^2+1)^(1/2)+x^2-ln(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{-\int \frac {2 \sqrt {x^{2}+1}\, x^{2}+x^{2}-\sqrt {x^{2}+1}}{\sqrt {x^{2}+1}\, x \left (\sqrt {x^{2}+1}+x^{2}-\ln \left (x \right )\right )}d x} \]

✓ Mathematica. Time used: 1.719 (sec). Leaf size: 94

ode=( x*y[x]/Sqrt[1+x^2] + 2*x*y[x] -y[x]/x  )+(  Sqrt[1+x^2] + x^2-Log[x] )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1 \exp \left (\int _1^x\frac {\sqrt {K[1]^2+1}-K[1]^2 \left (2 \sqrt {K[1]^2+1}+1\right )}{K[1] \left (\left (\sqrt {K[1]^2+1}+1\right ) K[1]^2-\sqrt {K[1]^2+1} \log (K[1])+1\right )}dK[1]\right ) \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 147.686 (sec). Leaf size: 121

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*y(x) + x*y(x)/sqrt(x**2 + 1) + (x**2 + sqrt(x**2 + 1) - log(x))*Derivative(y(x), x) - y(x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{- \int \frac {x}{x^{2} \sqrt {x^{2} + 1} + x^{2} - \sqrt {x^{2} + 1} \log {\left (x \right )} + 1}\, dx + \int \frac {\sqrt {x^{2} + 1}}{x \left (x^{2} \sqrt {x^{2} + 1} + x^{2} - \sqrt {x^{2} + 1} \log {\left (x \right )} + 1\right )}\, dx - 2 \int \frac {x \sqrt {x^{2} + 1}}{x^{2} \sqrt {x^{2} + 1} + x^{2} - \sqrt {x^{2} + 1} \log {\left (x \right )} + 1}\, dx} \]