56.29.4 problem Ex 5

Internal problem ID [14236]
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 : Thursday, October 02, 2025 at 09:27:02 AM
CAS classification : [[_3rd_order, _missing_y]]

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

\[ y = \frac {\left (9-12 x +16 c_2 \right ) {\mathrm e}^{2 x}}{32}-\frac {x^{3}}{12}-\frac {{\mathrm e}^{-2 x} c_1}{2}-\frac {x}{8}+c_3 \]

✓ Mathematica. Time used: 0.178 (sec). Leaf size: 49

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]
 

\begin{align*} y(x)&\to -\frac {x^3}{12}-\frac {x}{8}+\frac {1}{32} e^{2 x} (-12 x+9+16 c_1)-\frac {1}{2} c_2 e^{-2 x}+c_3 \end{align*}

✓ Sympy. Time used: 0.168 (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)
 

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