71.9.5 problem 5
Internal
problem
ID
[14484]
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
:
Tuesday, January 28, 2025 at 08:25:29 PM
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*}
✓ Solution by Maple
Time used: 6.767 (sec). Leaf size: 185
dsolve([sqrt(1-x)*diff(y(x),x$2)-4*y(x)=sin(x),y(-2) = 3, D(y)(-2) = -1],y(x), singsol=all)
\[
y = \frac {4 \pi \left (\left (\left (1-x \right )^{{3}/{2}}\right )^{{2}/{3}} \left (\left (\int _{-2}^{x}\frac {\operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \sqrt {\left (-\textit {\_z1} +1\right )^{{3}/{2}}}}{3}\right ) \sqrt {-\textit {\_z1} +1}\, \sin \left (\textit {\_z1} \right )}{\left (\left (-\textit {\_z1} +1\right )^{{3}/{2}}\right )^{{1}/{3}}}d \textit {\_z1} \right ) \sqrt {3}+6 \operatorname {BesselI}\left (-\frac {1}{3}, \frac {8 \,3^{{3}/{4}}}{3}\right ) 3^{{3}/{4}}-3 \operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \,3^{{3}/{4}}}{3}\right )\right ) \operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \sqrt {\left (1-x \right )^{{3}/{2}}}}{3}\right )+\operatorname {BesselI}\left (\frac {2}{3}, \frac {8 \sqrt {\left (1-x \right )^{{3}/{2}}}}{3}\right ) \left (x -1\right ) \left (6 \,3^{{3}/{4}} \operatorname {BesselI}\left (\frac {1}{3}, \frac {8 \,3^{{3}/{4}}}{3}\right )+\sqrt {3}\, \left (\int _{-2}^{x}\frac {\operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \sqrt {\left (-\textit {\_z1} +1\right )^{{3}/{2}}}}{3}\right ) \left (\left (-\textit {\_z1} +1\right )^{{3}/{2}}\right )^{{1}/{3}} \sin \left (\textit {\_z1} \right )}{\sqrt {-\textit {\_z1} +1}}d \textit {\_z1} \right )-3 \operatorname {BesselI}\left (-\frac {2}{3}, \frac {8 \,3^{{3}/{4}}}{3}\right )\right )\right )}{9 \left (\left (1-x \right )^{{3}/{2}}\right )^{{1}/{3}}}
\]
✗ Solution by Mathematica
Time used: 0.000 (sec). Leaf size: 0
DSolve[{Sqrt[1-x]*D[y[x],{x,2}]-4*y[x]==Sin[x],{y[-2]==3,Derivative[1][y][-2]==-1}},y[x],x,IncludeSingularSolutions -> True]
Not solved