51.1.69 problem 69

Internal problem ID [10339]
Book : First order enumerated odes
Section : section 1
Problem number : 69
Date solved : Tuesday, September 30, 2025 at 07:22:18 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} y^{\prime }&=x \,{\mathrm e}^{x +y}+\sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 29

ode:=diff(y(x),x) = x*exp(x+y(x))+sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\cos \left (x \right )-\ln \left (-c_1 -\int x \,{\mathrm e}^{x -\cos \left (x \right )}d x \right ) \]

✓ Mathematica. Time used: 0.282 (sec). Leaf size: 150

ode=D[y[x],x]==x*Exp[x+y[x]]+Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x\left (-\exp \left (K[2]-\int _1^{K[2]}-\sin (K[1])dK[1]\right ) K[2]-\exp \left (-y(x)-\int _1^{K[2]}-\sin (K[1])dK[1]\right ) \sin (K[2])\right )dK[2]+\int _1^{y(x)}-\exp \left (-K[3]-\int _1^x-\sin (K[1])dK[1]\right ) \left (\exp \left (K[3]+\int _1^x-\sin (K[1])dK[1]\right ) \int _1^x\exp \left (-K[3]-\int _1^{K[2]}-\sin (K[1])dK[1]\right ) \sin (K[2])dK[2]-1\right )dK[3]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 10.213 (sec). Leaf size: 24

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

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