62.29.10 problem Ex 12

Internal problem ID [12890]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 52. Summary. Page 117
Problem number : Ex 12
Date solved : Monday, March 31, 2025 at 07:23:43 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }-3 y^{\prime \prime }+5 y^{\prime }-2 y&={\mathrm e}^{3 x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 30

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)+5*diff(y(x),x)-2*y(x) = exp(3*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,{\mathrm e}^{-2 x}+\frac {{\mathrm e}^{3 x}}{40}+\left (c_4 \,x^{2}+c_3 x +c_1 \right ) {\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.08 (sec). Leaf size: 118

ode=D[y[x],{x,4}]-D[y[x],{x,3}]-3*D[y[x],{x,2}]+5*D[y[x],x]-2*y[x]==Exp[3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^x x \int _1^x-\frac {1}{9} e^{2 K[2]} (3 K[2]+1)dK[2]+e^x \int _1^x\frac {1}{54} e^{2 K[1]} \left (9 K[1]^2+6 K[1]+2\right )dK[1]+\frac {1}{12} e^{3 x} x^2+c_4 e^x x^2-\frac {e^{3 x}}{135}+c_3 e^x x+c_1 e^{-2 x}+c_2 e^x \]

✓ Sympy. Time used: 0.293 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{4} e^{- 2 x} + \left (C_{1} + x \left (C_{2} + C_{3} x\right )\right ) e^{x} + \frac {e^{3 x}}{40} \]