62.29.4 problem Ex 5

Internal problem ID [12876]
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 5
Date solved : Wednesday, March 05, 2025 at 08:49:26 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }-4 y^{\prime }&=x^{2}-3 \,{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 35

ode:=diff(diff(diff(y(x),x),x),x)-4*diff(y(x),x) = x^2-3*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (9-12 x +16 c_{1} \right ) {\mathrm e}^{2 x}}{32}-\frac {x^{3}}{12}-\frac {c_{2} {\mathrm e}^{-2 x}}{2}-\frac {x}{8}+c_3 \]

✓ Mathematica. Time used: 12.586 (sec). Leaf size: 99

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

\[ y(x)\to \int _1^xe^{-2 K[3]} \left (e^{4 K[3]} c_1+c_2+e^{4 K[3]} \int _1^{K[3]}\frac {1}{4} \left (e^{-2 K[1]} K[1]^2-3\right )dK[1]+\int _1^{K[3]}\left (\frac {3}{4} e^{4 K[2]}-\frac {1}{4} e^{2 K[2]} K[2]^2\right )dK[2]\right )dK[3]+c_3 \]

✓ Sympy. Time used: 0.285 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{1} + C_{3} e^{- 2 x} - \frac {x^{3}}{12} - \frac {x}{8} + \left (C_{2} - \frac {3 x}{8}\right ) e^{2 x} \]