22.2.70 problem 70

Internal problem ID [4513]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 70
Date solved : Tuesday, September 30, 2025 at 07:33:45 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 41

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

\[ y = \frac {7 c_2 \cos \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )+7 c_3 \sin \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )+x^{{3}/{2}} \left (x +7 c_1 \right )}{7 \sqrt {x}} \]

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 57

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

\begin{align*} y(x)&\to \frac {1}{7} x (x+7 c_1)+\frac {c_2 \cos \left (\frac {1}{2} \sqrt {3} \log (x)\right )}{\sqrt {x}}+\frac {c_3 \sin \left (\frac {1}{2} \sqrt {3} \log (x)\right )}{\sqrt {x}} \end{align*}

✓ Sympy. Time used: 0.290 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = \frac {C_{2} \sin {\left (\frac {\sqrt {3} \log {\left (x \right )}}{2} \right )} + C_{3} \cos {\left (\frac {\sqrt {3} \log {\left (x \right )}}{2} \right )} + \frac {x^{\frac {3}{2}} \left (C_{1} + x\right )}{7}}{\sqrt {x}} \]