6.10.35 problem 35

Internal problem ID [1839]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 35
Date solved : Tuesday, September 30, 2025 at 05:20:20 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} \left (x +1\right ) \left (2 x +3\right ) y^{\prime \prime }+2 \left (x +2\right ) y^{\prime }-2 y&=\left (2 x +3\right )^{2} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 20

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

\[ y = \frac {x^{2} \left (9+4 x \right )}{6 x +6} \]

✓ Mathematica. Time used: 1.935 (sec). Leaf size: 47

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

\begin{align*} y(x)&\to \frac {8 x^3+18 x^2-\frac {27 i \sqrt {2 x+3}}{\sqrt {-2 x-3}}+27}{12 x+12} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x + 1)*(2*x + 3)*Derivative(y(x), (x, 2)) - (2*x + 3)**2 + (2*x + 4)*Derivative(y(x), x) - 2*y(x),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-2*x**2*Derivative(y(x), (x, 2)) + 4*x**2 - 5*x*Derivative(y(x), (x, 2)) + 12*x + 2*y(x) - 3*Derivative(y(x), (x, 2)) + 9)/(2*(x + 2)) cannot be solved by the factorable group method