76.39.7 problem Ex. 7

Internal problem ID [20224]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VII. Linear equations with variable coefficients. End of chapter problems at page 91
Problem number : Ex. 7
Date solved : Friday, October 03, 2025 at 07:38:10 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} 16 \left (1+x \right )^{4} y^{\prime \prime \prime \prime }+96 \left (1+x \right )^{3} y^{\prime \prime \prime }+104 \left (1+x \right )^{2} y^{\prime \prime }+8 \left (1+x \right ) y^{\prime }+y&=x^{2}+4 x +3 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 48

ode:=16*(1+x)^4*diff(diff(diff(diff(y(x),x),x),x),x)+96*(1+x)^3*diff(diff(diff(y(x),x),x),x)+104*(1+x)^2*diff(diff(y(x),x),x)+8*(1+x)*diff(y(x),x)+y(x) = x^2+4*x+3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x^{2}+52 x +51\right ) \sqrt {x +1}+225 \left (c_4 x +c_3 +c_4 \right ) \ln \left (x +1\right )+225 c_2 x +225 c_1 +225 c_2}{225 \sqrt {x +1}} \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 61

ode=16*(x+1)^4*D[y[x],{x,4}]+96*(x+1)^3*D[y[x],{x,3}]+104*(x+1)^2*D[y[x],{x,2}]+8*(x+1)*D[y[x],x]+y[x]==x^2+4*x+3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{225} \left (x^2+52 x+\frac {225 c_1}{\sqrt {x+1}}+225 c_3 \sqrt {x+1}+51\right )+\frac {(c_4 (x+1)+c_2) \log (x+1)}{\sqrt {x+1}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 - 4*x + 16*(x + 1)**4*Derivative(y(x), (x, 4)) + 96*(x + 1)**3*Derivative(y(x), (x, 3)) + 104*(x + 1)**2*Derivative(y(x), (x, 2)) + (8*x + 8)*Derivative(y(x), x) + y(x) - 3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-16*x**4*Derivative(y(x), (x, 4)) - 96*x**3*Derivative(y(x), (x, 3)) - 64*x**3*Derivative(y(x), (x, 4)) - 104*x**2*Derivative(y(x), (x, 2)) - 288*x**2*Derivative(y(x), (x, 3)) - 96*x**2*Derivative(y(x), (x, 4)) + x**2 - 208*x*Derivative(y(x), (x, 2)) - 288*x*Derivative(y(x), (x, 3)) - 64*x*Derivative(y(x), (x, 4)) + 4*x - y(x) - 104*Derivative(y(x), (x, 2)) - 96*Derivative(y(x), (x, 3)) - 16*Derivative(y(x), (x, 4)) + 3)/(8*(x + 1)) cannot be solved by the factorable group method