52.2.7 problem 8

Internal problem ID [10430]
Book : Second order enumerated odes
Section : section 2
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 07:23:35 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.310 (sec). Leaf size: 882

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

\[ \text {Expression too large to display} \]

✓ Mathematica. Time used: 0.087 (sec). Leaf size: 175

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{x^4+4 K[2] x^3+6 K[2]^2 x^2+4 K[2]^3 x+K[2]^4+1}-\int _1^x-\frac {4 K[1]^3+12 K[2] K[1]^2+12 K[2]^2 K[1]+4 K[2]^3}{\left (K[1]^4+4 K[2] K[1]^3+6 K[2]^2 K[1]^2+4 K[2]^3 K[1]+K[2]^4+1\right )^2}dK[1]\right )dK[2]+\int _1^x\left (\frac {1}{K[1]^4+4 y(x) K[1]^3+6 y(x)^2 K[1]^2+4 y(x)^3 K[1]+y(x)^4+1}-1\right )dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.692 (sec). Leaf size: 109

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

\[ C_{1} + x - \frac {\sqrt {2} \log {\left (\left (- x - y{\left (x \right )}\right )^{2} - \sqrt {2} \left (- x - y{\left (x \right )}\right ) + 1 \right )}}{8} + \frac {\sqrt {2} \log {\left (\left (- x - y{\left (x \right )}\right )^{2} + \sqrt {2} \left (- x - y{\left (x \right )}\right ) + 1 \right )}}{8} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} \left (- x - y{\left (x \right )}\right ) - 1 \right )}}{4} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} \left (- x - y{\left (x \right )}\right ) + 1 \right )}}{4} = 0 \]