35.7.18 problem 21

Internal problem ID [6200]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations. page 435
Problem number : 21
Date solved : Sunday, March 30, 2025 at 10:42:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.55 (sec). Leaf size: 173

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

\[ y(x)\to \frac {1}{2} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (\left (\left (1+i \sqrt {3}\right ) x^{\frac {3}{2}+i \sqrt {3}}+\left (1-i \sqrt {3}\right ) x^{3/2}+2 c_1 x^{\frac {i \sqrt {3}}{2}}\right ) \cos \left (\frac {1}{2} \sqrt {3} \log (x)\right )+\left (\left (\sqrt {3}-i\right ) x^{\frac {3}{2}+i \sqrt {3}}+\left (\sqrt {3}+i\right ) x^{3/2}+2 c_2 x^{\frac {i \sqrt {3}}{2}}\right ) \sin \left (\frac {1}{2} \sqrt {3} \log (x)\right )\right ) \]

✓ Sympy. Time used: 0.257 (sec). Leaf size: 42

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

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