68.1.3 problem 4

Internal problem ID [17070]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 4
Date solved : Thursday, October 02, 2025 at 01:42:37 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 211

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

\[ y = {\mathrm e}^{-\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{{2}/{3}}-101 \left (404+60 \sqrt {69}\right )^{{2}/{3}}-484 \left (404+60 \sqrt {69}\right )^{{1}/{3}}-3872\right ) x}{5808}} \left (c_2 \cos \left (\frac {\left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}} \sqrt {3}\, \left (15 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {23}-101 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}}+484\right ) x}{5808}\right )+c_3 \sin \left (\frac {\left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}} \sqrt {3}\, \left (15 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {23}-101 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}}+484\right ) x}{5808}\right )\right )+c_1 \,{\mathrm e}^{\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{{2}/{3}}-101 \left (404+60 \sqrt {69}\right )^{{2}/{3}}-484 \left (404+60 \sqrt {69}\right )^{{1}/{3}}+1936\right ) x}{2904}}+\frac {{\mathrm e}^{x}}{5} \]

✓ Mathematica. Time used: 0.243 (sec). Leaf size: 504

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

\begin{align*} y(x)&\to \exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ]\right ) \int _1^x\frac {i \exp \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ] K[3]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ] K[3]-K[3]\right ) \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ]\right )}{5 \sqrt {23}}dK[3]+\exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ]\right ) \int _1^x-\frac {i \exp \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ] K[2]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ] K[2]-K[2]\right ) \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ]\right )}{5 \sqrt {23}}dK[2]+\exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ]\right ) \int _1^x\frac {i \exp \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ] K[1]+\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ] K[1]-K[1]\right ) \left (\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ]\right )}{5 \sqrt {23}}dK[1]+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,1\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+5 \text {$\#$1}+1\&,3\right ]\right ) \end{align*}

✓ Sympy. Time used: 0.385 (sec). Leaf size: 231

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

\[ y{\left (x \right )} = C_{1} e^{\frac {x \left (- \frac {22 \sqrt [3]{2}}{\sqrt [3]{101 + 15 \sqrt {69}}} + 8 + 2^{\frac {2}{3}} \sqrt [3]{101 + 15 \sqrt {69}}\right )}{12}} \sin {\left (\frac {\sqrt [3]{2} \sqrt {3} x \left (\frac {22}{\sqrt [3]{101 + 15 \sqrt {69}}} + \sqrt [3]{2} \sqrt [3]{101 + 15 \sqrt {69}}\right )}{12} \right )} + C_{2} e^{\frac {x \left (- \frac {22 \sqrt [3]{2}}{\sqrt [3]{101 + 15 \sqrt {69}}} + 8 + 2^{\frac {2}{3}} \sqrt [3]{101 + 15 \sqrt {69}}\right )}{12}} \cos {\left (\frac {\sqrt [3]{2} \sqrt {3} x \left (\frac {22}{\sqrt [3]{101 + 15 \sqrt {69}}} + \sqrt [3]{2} \sqrt [3]{101 + 15 \sqrt {69}}\right )}{12} \right )} + C_{3} e^{\frac {x \left (- 2^{\frac {2}{3}} \sqrt [3]{101 + 15 \sqrt {69}} + 4 + \frac {22 \sqrt [3]{2}}{\sqrt [3]{101 + 15 \sqrt {69}}}\right )}{6}} + \frac {e^{x}}{5} \]