83.36.11 problem 11

Internal problem ID [19374]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (A) at page 125
Problem number : 11
Date solved : Monday, March 31, 2025 at 07:11:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{3}-a \right ) y^{\prime \prime }-6 x^{2} y^{\prime }+6 x y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 15

ode:=(2*x^3-a)*diff(diff(y(x),x),x)-6*x^2*diff(y(x),x)+6*x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 x +c_2 \left (x^{3}+a \right ) \]

✓ Mathematica. Time used: 0.437 (sec). Leaf size: 22

ode=(2*x^3-a)*D[y[x],{x,2}]-6*x^2*D[y[x],x]+6*x*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -a c_2-c_2 x^3+c_1 x \]

✓ Sympy. Time used: 0.596 (sec). Leaf size: 82

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-6*x**2*Derivative(y(x), x) + 6*x*y(x) + (-a + 2*x**3)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\left (- a + 2 x^{3}\right )^{\frac {4}{3}} \left (C_{1} \sqrt [3]{\frac {x^{3}}{- a + 2 x^{3}}} {{}_{1}F_{0}\left (\begin {matrix} - \frac {2}{3} \\ \end {matrix}\middle | {\frac {2 x^{3}}{- a + 2 x^{3}}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} -1, 1 \\ \frac {2}{3} \end {matrix}\middle | {\frac {2 x^{3}}{- a + 2 x^{3}}} \right )}\right ) \sqrt [3]{x^{3}}}{x \sqrt [3]{- \frac {a}{2} + x^{3}}} \]