78.2.34 problem 10.d

Internal problem ID [20986]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 2, Second order ODEs. Problems section 2.6
Problem number : 10.d
Date solved : Thursday, October 02, 2025 at 07:01:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }+3 y&=5 x^{2} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=3 \\ y^{\prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 39

ode:=x^2*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+3*y(x) = 5*x^2; 
ic:=[y(1) = 3, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {14 x^{{3}/{2}} \sin \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right ) \sqrt {3}}{3}-2 x^{{3}/{2}} \cos \left (\frac {\sqrt {3}\, \ln \left (x \right )}{2}\right )+5 x^{2} \]

✓ Mathematica. Time used: 0.087 (sec). Leaf size: 54

ode=x^2*D[y[x],{x,2}]-2*x*D[y[x],x]+3*y[x]==5*x^2; 
ic={y[1]==3,Derivative[1][y][1] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {14 x^{3/2} \sin \left (\frac {1}{2} \sqrt {3} \log (x)\right )}{\sqrt {3}}-2 x^{3/2} \cos \left (\frac {1}{2} \sqrt {3} \log (x)\right )+5 x^2 \end{align*}

✓ Sympy. Time used: 0.216 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 5*x**2 - 2*x*Derivative(y(x), x) + 3*y(x),0) 
ics = {y(1): 3, Subs(Derivative(y(x), x), x, 1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {14 \sqrt {3} x^{\frac {3}{2}} \sin {\left (\frac {\sqrt {3} \log {\left (x \right )}}{2} \right )}}{3} - 2 x^{\frac {3}{2}} \cos {\left (\frac {\sqrt {3} \log {\left (x \right )}}{2} \right )} + 5 x^{2} \]