81.2.9 problem 9

Internal problem ID [18617]
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 : Tuesday, January 28, 2025 at 12:04:18 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*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 166

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 69

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

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