65.9.5 problem 5

Internal problem ID [15771]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.1, page 186
Problem number : 5
Date solved : Friday, October 03, 2025 at 07:30:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \sqrt {1-x}\, y^{\prime \prime }-4 y&=\sin \left (x \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.984 (sec). Leaf size: 185

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

\[ y = \frac {4 \left (\left (\int _{-2}^{x}\frac {\operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \sqrt {\left (1-\textit {\_z1} \right )^{{3}/{2}}}}{3}\right ) \sqrt {1-\textit {\_z1}}\, \sin \left (\textit {\_z1} \right )}{\left (\left (1-\textit {\_z1} \right )^{{3}/{2}}\right )^{{1}/{3}}}d \textit {\_z1} \sqrt {3}+6 \operatorname {BesselI}\left (-\frac {1}{3}, \frac {8 \,3^{{3}/{4}}}{3}\right ) 3^{{3}/{4}}-3 \operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \,3^{{3}/{4}}}{3}\right )\right ) \left (\left (1-x \right )^{{3}/{2}}\right )^{{2}/{3}} \operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \sqrt {\left (1-x \right )^{{3}/{2}}}}{3}\right )+\left (6 \operatorname {BesselI}\left (\frac {1}{3}, \frac {8 \,3^{{3}/{4}}}{3}\right ) 3^{{3}/{4}}+\int _{-2}^{x}\frac {\operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \sqrt {\left (1-\textit {\_z1} \right )^{{3}/{2}}}}{3}\right ) \left (\left (1-\textit {\_z1} \right )^{{3}/{2}}\right )^{{1}/{3}} \sin \left (\textit {\_z1} \right )}{\sqrt {1-\textit {\_z1}}}d \textit {\_z1} \sqrt {3}-3 \operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \,3^{{3}/{4}}}{3}\right )\right ) \left (-1+x \right ) \operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \sqrt {\left (1-x \right )^{{3}/{2}}}}{3}\right )\right ) \pi }{9 \left (\left (1-x \right )^{{3}/{2}}\right )^{{1}/{3}}} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : solve: Cannot solve sqrt(1 - x)*Derivative(y(x), (x, 2)) - 4*y(x) - sin(x)