83.33.3 problem 3

Internal problem ID [19340]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (G) at page 115
Problem number : 3
Date solved : Monday, March 31, 2025 at 07:09:01 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (5\right )}-n^{2} y^{\prime \prime \prime }&={\mathrm e}^{a x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 101

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-n^2*diff(diff(diff(y(x),x),x),x) = exp(a*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {2 c_2 \left (-a^{5}+a^{3} n^{2}\right ) {\mathrm e}^{-n x}+2 c_1 \left (a^{5}-a^{3} n^{2}\right ) {\mathrm e}^{n x}+n^{3} \left (2 \,{\mathrm e}^{a x}+a^{3} \left (a -n \right ) \left (a +n \right ) \left (c_3 \,x^{2}+2 c_4 x +2 c_5 \right )\right )}{2 a^{5} n^{3}-2 a^{3} n^{5}} \]

✓ Mathematica. Time used: 0.632 (sec). Leaf size: 73

ode=D[y[x],{x,5}]-n^2*D[y[x],{x,3}]==Exp[a*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {\frac {e^{a x}}{a^3}+\frac {(a-n) (a+n) \left (c_1 e^{n x}-c_2 e^{-n x}\right )}{n^3}}{(n-a) (a+n)}+c_5 x^2+c_4 x+c_3 \]

✓ Sympy. Time used: 0.154 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-n**2*Derivative(y(x), (x, 3)) - exp(a*x) + Derivative(y(x), (x, 5)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{2} x + C_{3} x^{2} + C_{4} e^{- n x} + C_{5} e^{n x} + \frac {e^{a x}}{a^{3} \left (a^{2} - n^{2}\right )} \]