62.27.6 problem Ex 6

Internal problem ID [12865]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 50. Method of undetermined coefficients. Page 107
Problem number : Ex 6
Date solved : Wednesday, March 05, 2025 at 08:48:53 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }-3 y^{\prime }&=3 x^{2}+\sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 41

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

\[ y = -\frac {x^{3}}{3}+\frac {2 x^{2}}{3}-c_{2} {\mathrm e}^{-x}+\frac {{\mathrm e}^{3 x} c_{1}}{3}+\frac {\sin \left (x \right )}{10}+\frac {\cos \left (x \right )}{5}-\frac {14 x}{9}+c_3 \]

✓ Mathematica. Time used: 27.194 (sec). Leaf size: 95

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

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

✓ Sympy. Time used: 0.255 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{1} + C_{2} e^{- x} + C_{3} e^{3 x} - \frac {x^{3}}{3} + \frac {2 x^{2}}{3} - \frac {14 x}{9} + \frac {\sin {\left (x \right )}}{10} + \frac {\cos {\left (x \right )}}{5} \]