83.37.5 problem 5

Internal problem ID [19371]
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 (B) at page 128
Problem number : 5
Date solved : Thursday, March 13, 2025 at 02:21:20 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 137

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

\[ y \left (x \right ) = \frac {4 \,{\mathrm e}^{\frac {x \left (b x -2 \sqrt {-b}\right )}{2}} c_{1} b^{{3}/{2}}+4 \,{\mathrm e}^{\frac {x \left (b x +2 \sqrt {-b}\right )}{2}} c_{2} b^{{3}/{2}}-\operatorname {erf}\left (\frac {\sqrt {2}\, \left (b x +\sqrt {-b}\right )}{2 \sqrt {b}}\right ) \sqrt {\pi }\, \sqrt {2}\, {\mathrm e}^{-\frac {1}{2}+\frac {b \,x^{2}}{2}+x \sqrt {-b}}+{\mathrm e}^{-\frac {1}{2}+\frac {b \,x^{2}}{2}-x \sqrt {-b}} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (-b x +\sqrt {-b}\right )}{2 \sqrt {b}}\right )}{4 b^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.457 (sec). Leaf size: 139

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

\[ y(x)\to \frac {e^{\frac {1}{2} \left (\sqrt {b} x-i\right )^2} \left (-\sqrt {2 \pi } e^{2 i \sqrt {b} x} \text {erf}\left (\frac {\sqrt {b} x+i}{\sqrt {2}}\right )+i \sqrt {2 \pi } \text {erfi}\left (\frac {1+i \sqrt {b} x}{\sqrt {2}}\right )+2 \sqrt {e} b \left (2 \sqrt {b} c_1-i c_2 e^{2 i \sqrt {b} x}\right )\right )}{4 b^{3/2}} \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (b**2*x**2*y(x) - x + Derivative(y(x), (x, 2)))/(2*b*x) cannot be solved by the factorable group method