51.1.63 problem 63

Internal problem ID [10333]
Book : First order enumerated odes
Section : section 1
Problem number : 63
Date solved : Tuesday, September 30, 2025 at 07:19:39 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 49

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

\[ y = -b x +\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a}^{4}+4 \textit {\_a}^{3} a +6 \textit {\_a}^{2} a^{2}+4 \textit {\_a} \,a^{3}+a^{4}+b}d \textit {\_a} +c_1 \right ) \]

✓ Mathematica. Time used: 0.3 (sec). Leaf size: 523

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\int _1^x\frac {b \left (4 a^3+12 b K[1] a^2+12 K[2] a^2+12 b^2 K[1]^2 a+12 K[2]^2 a+24 b K[1] K[2] a+4 b^3 K[1]^3+4 K[2]^3+12 b K[1] K[2]^2+12 b^2 K[1]^2 K[2]\right )}{\left (a^4+4 b K[1] a^3+4 K[2] a^3+6 b^2 K[1]^2 a^2+6 K[2]^2 a^2+12 b K[1] K[2] a^2+4 b^3 K[1]^3 a+4 K[2]^3 a+12 b K[1] K[2]^2 a+12 b^2 K[1]^2 K[2] a+b^4 K[1]^4+K[2]^4+4 b K[1] K[2]^3+6 b^2 K[1]^2 K[2]^2+b+4 b^3 K[1]^3 K[2]\right )^2}dK[1]-\frac {1}{a^4+4 b x a^3+4 K[2] a^3+6 b^2 x^2 a^2+6 K[2]^2 a^2+12 b x K[2] a^2+4 b^3 x^3 a+4 K[2]^3 a+12 b x K[2]^2 a+12 b^2 x^2 K[2] a+b^4 x^4+K[2]^4+4 b x K[2]^3+6 b^2 x^2 K[2]^2+b+4 b^3 x^3 K[2]}\right )dK[2]+\int _1^x\left (1-\frac {b}{a^4+4 b K[1] a^3+4 y(x) a^3+6 b^2 K[1]^2 a^2+6 y(x)^2 a^2+12 b K[1] y(x) a^2+4 b^3 K[1]^3 a+4 y(x)^3 a+12 b K[1] y(x)^2 a+12 b^2 K[1]^2 y(x) a+b^4 K[1]^4+y(x)^4+4 b K[1] y(x)^3+6 b^2 K[1]^2 y(x)^2+b+4 b^3 K[1]^3 y(x)}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

TypeError : argument of type Mul is not iterable