87.13.19 problem 20

Internal problem ID [23502]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 100
Problem number : 20
Date solved : Thursday, October 02, 2025 at 09:42:32 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} 8 y-8 x y^{\prime }+4 x^{2} y^{\prime \prime }+x^{3} y^{\prime \prime \prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ y^{\prime }\left (1\right )&=1 \\ y^{\prime \prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 18

ode:=x^3*diff(diff(diff(y(x),x),x),x)+4*x^2*diff(diff(y(x),x),x)-8*x*diff(y(x),x)+8*y(x) = 0; 
ic:=[y(1) = 0, D(y)(1) = 1, (D@@2)(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {3 x}{5}+\frac {2 x^{2}}{3}-\frac {1}{15 x^{4}} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 24

ode=x^3*D[y[x],{x,3}]+4*x^2*D[y[x],{x,2}]-8*x*D[y[x],x]+8*y[x]==0; 
ic={y[1]==0,Derivative[1][y][1] ==1,Derivative[2][y][1] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {-10 x^6+9 x^5+1}{15 x^4} \end{align*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \frac {5 x^{2}}{6} - \frac {4 x}{5} - \frac {1}{30 x^{4}} \]