60.2.175 problem 751

Internal problem ID [10749]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 751
Date solved : Wednesday, March 05, 2025 at 12:34:10 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }&=\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )+x^{4}\right ) y}{x \left (x +1\right )} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 24

ode:=diff(y(x),x) = (ln(y(x))*x+ln(y(x))+x^4)*y(x)/x/(1+x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x +1\right )^{x} {\mathrm e}^{\frac {x \left (x^{2}+2 c_{1} -2 x \right )}{2}} \]

✓ Mathematica. Time used: 0.208 (sec). Leaf size: 73

ode=D[y[x],x] == ((x^4 + Log[y[x]] + x*Log[y[x]])*y[x])/(x*(1 + x)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{x K[2]}-\int _1^x-\frac {1}{K[1]^2 K[2]}dK[1]\right )dK[2]+\int _1^x\left (-K[1]-\frac {1}{K[1]+1}+1-\frac {\log (y(x))}{K[1]^2}\right )dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.998 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**4 + x*log(y(x)) + log(y(x)))*y(x)/(x*(x + 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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