75.6.29 problem 162

Internal problem ID [16714]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 6. Linear equations of the first order. The Bernoulli equation. Exercises page 54
Problem number : 162
Date solved : Monday, March 31, 2025 at 03:07:20 PM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 36

ode:=2*diff(y(x),x)*ln(x)+y(x)/x = cos(x)/y(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {\ln \left (x \right ) \left (\sin \left (x \right )+c_1 \right )}}{\ln \left (x \right )} \\ y &= -\frac {\sqrt {\ln \left (x \right ) \left (\sin \left (x \right )+c_1 \right )}}{\ln \left (x \right )} \\ \end{align*}

✓ Mathematica. Time used: 0.345 (sec). Leaf size: 70

ode=2*D[y[x],x]*Log[x]+y[x]/x==Cos[x]/y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.694 (sec). Leaf size: 29

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

\[ \left [ y{\left (x \right )} = - \sqrt {\frac {C_{1} + \sin {\left (x \right )}}{\log {\left (x \right )}}}, \ y{\left (x \right )} = \sqrt {\frac {C_{1} + \sin {\left (x \right )}}{\log {\left (x \right )}}}\right ] \]