75.2.9 problem 9

Internal problem ID [19903]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter II. Change of variable. Exercises at page 20
Problem number : 9
Date solved : Thursday, October 02, 2025 at 05:00:44 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} v^{\prime \prime \prime }+2 x^{2} v^{\prime \prime }+v&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 184

ode:=x^3*diff(diff(diff(v(x),x),x),x)+2*x^2*diff(diff(v(x),x),x)+v(x) = 0; 
dsolve(ode,v(x), singsol=all);
 

\[ v = c_1 \,x^{-\frac {\left (100+12 \sqrt {69}\right )^{{2}/{3}}-2 \left (100+12 \sqrt {69}\right )^{{1}/{3}}+4}{6 \left (100+12 \sqrt {69}\right )^{{1}/{3}}}}+c_2 \,x^{\frac {4+\left (100+12 \sqrt {69}\right )^{{2}/{3}}+4 \left (100+12 \sqrt {69}\right )^{{1}/{3}}}{12 \left (100+12 \sqrt {69}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (100+12 \sqrt {3}\, \sqrt {23}\right )^{{2}/{3}}-4\right ) \ln \left (x \right )}{12 \left (100+12 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}}}\right )+c_3 \,x^{\frac {4+\left (100+12 \sqrt {69}\right )^{{2}/{3}}+4 \left (100+12 \sqrt {69}\right )^{{1}/{3}}}{12 \left (100+12 \sqrt {69}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (100+12 \sqrt {3}\, \sqrt {23}\right )^{{2}/{3}}-4\right ) \ln \left (x \right )}{12 \left (100+12 \sqrt {3}\, \sqrt {23}\right )^{{1}/{3}}}\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 69

ode=x^3*D[v[x],{x,3}]+2*x^2*D[v[x],{x,2}]+v[x]==0; 
ic={}; 
DSolve[{ode,ic},v[x],x,IncludeSingularSolutions->True]
 

\begin{align*} v(x)&\to c_3 x^{\text {Root}\left [\text {$\#$1}^3-\text {$\#$1}^2+1\&,3\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^3-\text {$\#$1}^2+1\&,2\right ]}+c_1 x^{\text {Root}\left [\text {$\#$1}^3-\text {$\#$1}^2+1\&,1\right ]} \end{align*}

✓ Sympy. Time used: 0.072 (sec). Leaf size: 39

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

\[ v{\left (x \right )} = \frac {C_{1}}{x^{- \operatorname {CRootOf} {\left (x^{3} - x^{2} + 1, 0\right )}}} + C_{2} x^{\operatorname {CRootOf} {\left (x^{3} - x^{2} + 1, 1\right )}} + C_{3} x^{\operatorname {CRootOf} {\left (x^{3} - x^{2} + 1, 2\right )}} \]