68.15.41 problem 41

Internal problem ID [17767]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.7, page 195
Problem number : 41
Date solved : Thursday, October 02, 2025 at 02:27:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+y&=x^{3} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 21

ode:=4*x^2*diff(diff(y(x),x),x)+y(x) = x^3; 
ic:=[y(1) = 1, D(y)(1) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {8 \left (3-5 \ln \left (x \right )\right ) \sqrt {x}}{25}+\frac {x^{3}}{25} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 25

ode=4*x^2*D[y[x],{x,2}]+y[x]==x^3; 
ic={y[1]==1,Derivative[1][y][1]==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{25} \sqrt {x} \left (x^{5/2}-40 \log (x)+24\right ) \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = - \frac {8 \sqrt {x} \log {\left (x \right )}}{5} + \frac {24 \sqrt {x}}{25} + \frac {x^{3}}{25} \]