56.4.13 problem 13
Internal
problem
ID
[8902]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
13
Date
solved
:
Monday, January 27, 2025 at 05:18:56 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+\left (\cos \left (x \right )-1\right ) y^{\prime }+{\mathrm e}^{x} y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.169 (sec). Leaf size: 300
Order:=6;
dsolve(x^2*diff(y(x), x$2) + (cos(x)-1)*diff(y(x), x) + exp(x)*y(x) = 0,y(x),type='series',x=0);
\[
y = \sqrt {x}\, \left (c_{2} x^{\frac {i \sqrt {3}}{2}} \left (1+\frac {1}{4} i \sqrt {3} x +\frac {-i \sqrt {3}-11}{32 i \sqrt {3}+64} x^{2}+\frac {\frac {55 \sqrt {3}}{288}+\frac {55 i}{96}}{\left (i-\sqrt {3}\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right )} x^{3}+\frac {1}{384} \frac {112 i \sqrt {3}+199}{\left (-\sqrt {3}+2 i\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (-i+\sqrt {3}\right )} x^{4}+\frac {\frac {18491 \sqrt {3}}{38400}+\frac {4387 i}{12800}}{\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+3\right ) \left (i \sqrt {3}+2\right ) \left (-i+\sqrt {3}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{4} i \sqrt {3} x +\frac {-\sqrt {3}-11 i}{64 i+32 \sqrt {3}} x^{2}+\frac {55 \sqrt {3}-165 i}{3456 i-2304 \sqrt {3}} x^{3}+\frac {199 i+112 \sqrt {3}}{-27648 i+7680 \sqrt {3}} x^{4}+\frac {\frac {18491 \sqrt {3}}{38400}-\frac {4387 i}{12800}}{\left (2 i+\sqrt {3}\right ) \left (\sqrt {3}+5 i\right ) \left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+3 i\right ) \left (\sqrt {3}+i\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 2502
AsymptoticDSolveValue[x^2*D[y[x],{x,2}] + (Cos[x]-1)*D[y[x],x] + Exp[x]*y[x] ==0,y[x],{x,0,"6"-1}]
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