12.6.2 problem 2
Internal
problem
ID
[1681]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
Exact
equations.
Section
2.5
Page
79
Problem
number
:
2
Date
solved
:
Monday, January 27, 2025 at 05:25:23 AM
CAS
classification
:
[_linear]
\begin{align*} 3 y \cos \left (x \right )+4 x \,{\mathrm e}^{x}+2 x^{3} y+\left (3 \sin \left (x \right )+3\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 329
dsolve((3*y(x)*cos(x)+4*x*exp(x)+2*x^3*y(x))+(3*sin(x)+3)*diff(y(x),x)=0,y(x), singsol=all)
\[
y = -\frac {{\mathrm e}^{\frac {\left (24 i x \,{\mathrm e}^{i x}-24 x \right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (-24 i-24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )+4 i x \left (x^{2}+\frac {3}{4}\right ) {\mathrm e}^{i x}-3 x}{3 \,{\mathrm e}^{i x}+3 i}} \left (4 \left (\int \frac {x \left (1-i {\mathrm e}^{i x}\right )^{4 x^{2}} \left ({\mathrm e}^{\frac {\left (24 i+24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-4 x \left (\left (6 i {\mathrm e}^{i x}-6\right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (i x^{2}-\frac {3}{4}-\frac {3}{4} i\right ) {\mathrm e}^{i x}+\frac {3}{4}-\frac {3 i}{4}\right )}{3 \,{\mathrm e}^{i x}+3 i}}+2 i {\mathrm e}^{\frac {24 \left ({\mathrm e}^{i x}+i\right ) x \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+4 \,{\mathrm e}^{i x} x^{3}+3 i x \,{\mathrm e}^{i x}+24 i {\mathrm e}^{i x} \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-3 x -24 \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )}{3 i {\mathrm e}^{i x}-3}}-{\mathrm e}^{\frac {\left (24 i+24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-4 \left (\left (6 i {\mathrm e}^{i x}-6\right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (i x^{2}-\frac {3}{4}+\frac {3}{4} i\right ) {\mathrm e}^{i x}-\frac {3}{4}-\frac {3 i}{4}\right ) x}{3 \,{\mathrm e}^{i x}+3 i}}\right )}{\sin \left (x \right )+1}d x \right )-3 c_1 \right ) \left (1-i {\mathrm e}^{i x}\right )^{-4 x^{2}}}{3 \left ({\mathrm e}^{i x}+i\right )^{2}}
\]
✓ Solution by Mathematica
Time used: 21.366 (sec). Leaf size: 193
DSolve[(3*y[x]*Cos[x]+4*x*Exp[x]+2*x^3*y[x])+(3*Sin[x]+3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {\left (1+i e^{-i x}\right )^{-4 x^2} \exp \left (-8 i x \operatorname {PolyLog}\left (2,-i e^{-i x}\right )-8 \operatorname {PolyLog}\left (3,-i e^{-i x}\right )+\frac {2}{3} x^3 \left (\cot \left (\frac {1}{4} (2 x+\pi )\right )-i\right )\right ) \left (\int _1^x-\frac {4}{3} \exp \left (-\frac {2}{3} \cot \left (\frac {1}{4} (2 K[1]+\pi )\right ) K[1]^3+\frac {2}{3} i K[1]^3+8 i \operatorname {PolyLog}\left (2,-i e^{-i K[1]}\right ) K[1]+K[1]+8 \operatorname {PolyLog}\left (3,-i e^{-i K[1]}\right )\right ) K[1] (i \cos (K[1])+\sin (K[1])+1)^{4 K[1]^2}dK[1]+c_1\right )}{\sin (x)+1}
\]