88.20.10 problem 10

Internal problem ID [24148]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 146
Problem number : 10
Date solved : Thursday, October 02, 2025 at 10:00:13 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (8\right )}+8 y^{\left (7\right )}+28 y^{\left (6\right )}+56 y^{\left (5\right )}+70 y^{\prime \prime \prime \prime }+56 y^{\prime \prime \prime }+28 y^{\prime \prime }+8 y^{\prime }&={\mathrm e}^{-x} x^{9} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 99

ode:=diff(diff(diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x),x),x)+8*diff(diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x),x)+28*diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)+56*diff(diff(diff(diff(diff(y(x),x),x),x),x),x)+70*diff(diff(diff(diff(y(x),x),x),x),x)+56*diff(diff(diff(y(x),x),x),x)+28*diff(diff(y(x),x),x)+8*diff(y(x),x) = exp(-x)*x^9; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \int \left (\left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) c_4 +c_5 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (2+\sqrt {2}\right ) x}{2}}+\left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) c_6 +\sin \left (\frac {\sqrt {2}\, x}{2}\right ) c_7 \right ) {\mathrm e}^{\frac {\left (\sqrt {2}-2\right ) x}{2}}+\left (x^{9}-9 x^{8}+\sin \left (x \right ) c_3 +c_2 \cos \left (x \right )+362880 x -362880\right ) {\mathrm e}^{-x}+c_1 \,{\mathrm e}^{-2 x}\right )d x +c_8 \]

✓ Mathematica. Time used: 0.95 (sec). Leaf size: 294

ode=D[y[x],{x,8}]+8*D[y[x],{x,7}]+28*D[y[x],{x,6}]+56*D[y[x],{x,5}]+70*D[y[x],{x,4}]+56*D[y[x],{x,3}]+28*D[y[x],{x,2}]+8*D[y[x],x]==Exp[-x]*x^9; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_8-\frac {1}{2} e^{-\frac {1}{2} \left (4+\sqrt {2}\right ) x} \left (2 e^{\frac {x}{\sqrt {2}}+x} x^9+725760 e^{\frac {x}{\sqrt {2}}+x} x+c_7 e^{\frac {x}{\sqrt {2}}}+(c_5+c_6) e^{\frac {x}{\sqrt {2}}+x} \cos (x)+\left (\left (c_2+\left (\sqrt {2}-1\right ) c_3\right ) e^x+\left (\left (1+\sqrt {2}\right ) c_1+c_4\right ) e^{\left (1+\sqrt {2}\right ) x}\right ) \cos \left (\frac {x}{\sqrt {2}}\right )+c_5 e^{\frac {x}{\sqrt {2}}+x} \sin (x)-c_6 e^{\frac {x}{\sqrt {2}}+x} \sin (x)+c_1 e^{\left (1+\sqrt {2}\right ) x} \sin \left (\frac {x}{\sqrt {2}}\right )-\sqrt {2} c_2 e^x \sin \left (\frac {x}{\sqrt {2}}\right )+c_2 e^x \sin \left (\frac {x}{\sqrt {2}}\right )+c_3 e^x \sin \left (\frac {x}{\sqrt {2}}\right )-\sqrt {2} c_4 e^{\left (1+\sqrt {2}\right ) x} \sin \left (\frac {x}{\sqrt {2}}\right )-c_4 e^{\left (1+\sqrt {2}\right ) x} \sin \left (\frac {x}{\sqrt {2}}\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**9*exp(-x) + 8*Derivative(y(x), x) + 28*Derivative(y(x), (x, 2)) + 56*Derivative(y(x), (x, 3)) + 70*Derivative(y(x), (x, 4)) + 56*Derivative(y(x), (x, 5)) + 28*Derivative(y(x), (x, 6)) + 8*Derivative(y(x), (x, 7)) + Derivative(y(x), (x, 8)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(x**9 - (28*Derivative(y(x), (x, 2)) + 56*Derivative(y(x), (x,