54.2.419 problem 998

Internal problem ID [12293]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 998
Date solved : Wednesday, October 01, 2025 at 01:37:43 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 27

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

\[ y = \ln \left (x \right )+\operatorname {Ci}\left (x \right )+\frac {-c_1 \,x^{2}+1}{c_1 \,x^{2}+1} \]

✓ Mathematica. Time used: 0.319 (sec). Leaf size: 36

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

\begin{align*} y(x)&\to \operatorname {CosIntegral}(x)-\frac {2 x^2}{x^2-2 c_1}+\log (x)+1\\ y(x)&\to \operatorname {CosIntegral}(x)+\log (x)+1 \end{align*}

✓ Sympy. Time used: 1.374 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = \frac {C_{1} x^{2} \log {\left (x \right )} + C_{1} x^{2} \operatorname {Ci}{\left (x \right )} - C_{1} x^{2} - \log {\left (x \right )} - \operatorname {Ci}{\left (x \right )} - 1}{C_{1} x^{2} - 1} \]