Internal problem ID [1031]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_linear]
Solve \begin {gather*} \boxed {3 \cos \relax (x ) y+4 \,{\mathrm e}^{x} x +2 y x^{3}+\left (3 \sin \relax (x )+3\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.038 (sec). Leaf size: 490
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)
\[ y \relax (x ) = \frac {\left (1-i {\mathrm e}^{i x}\right )^{-4 x^{2}} {\mathrm e}^{\frac {4 i x^{3} {\mathrm e}^{i x}+24 i x \polylog \left (2, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 x \polylog \left (2, i {\mathrm e}^{i x}\right )+3 i x \,{\mathrm e}^{i x}-3 x -24 \polylog \left (3, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 i \polylog \left (3, i {\mathrm e}^{i x}\right )}{3 \,{\mathrm e}^{i x}+3 i}} \left (3 c_{1}-4 \left (\int \frac {x \left (1-i {\mathrm e}^{i x}\right )^{4 x^{2}} \left ({\mathrm e}^{-\frac {4 i x^{3} {\mathrm e}^{i x}+24 i x \polylog \left (2, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-3 i x \,{\mathrm e}^{i x}+3 x -24 x \polylog \left (2, i {\mathrm e}^{i x}\right )-24 \polylog \left (3, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 i \polylog \left (3, i {\mathrm e}^{i x}\right )-3 x \,{\mathrm e}^{i x}-3 i x}{3 \left ({\mathrm e}^{i x}+i\right )}}+2 i {\mathrm e}^{-\frac {4 i x^{3} {\mathrm e}^{i x}+24 i x \polylog \left (2, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 x \polylog \left (2, i {\mathrm e}^{i x}\right )-24 \polylog \left (3, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 i \polylog \left (3, i {\mathrm e}^{i x}\right )-3 x \,{\mathrm e}^{i x}-3 i x}{3 \left ({\mathrm e}^{i x}+i\right )}}-{\mathrm e}^{-\frac {4 i x^{3} {\mathrm e}^{i x}+24 i x \polylog \left (2, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 x \polylog \left (2, i {\mathrm e}^{i x}\right )+3 i x \,{\mathrm e}^{i x}-3 x -24 \polylog \left (3, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-3 x \,{\mathrm e}^{i x}-24 i \polylog \left (3, i {\mathrm e}^{i x}\right )-3 i x}{3 \left ({\mathrm e}^{i x}+i\right )}}\right )}{\sin \relax (x )+1}d x \right )\right )}{3 \left ({\mathrm e}^{i x}+i\right )^{2}} \]
✓ Solution by Mathematica
Time used: 65.575 (sec). Leaf size: 195
DSolve[(3*y[x]*Cos[x]+4*x*Exp[x]+2*x^3*y[x])+(3*Sin[x]+3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (\sin (x)-i \cos (x)+1)^{-4 x^2-2} \exp \left (8 i x \text {PolyLog}\left (2,i e^{i x}\right )-8 \text {PolyLog}\left (3,i e^{i x}\right )+\frac {1}{3} x \left (2 i x^2+\frac {2 x^2 \cos (x)}{\sin (x)+1}+3 i\right )\right ) \left (\int _1^x\frac {8}{3} i \exp \left (-\frac {2 \cos (K[1]) K[1]^3}{3 (\sin (K[1])+1)}-\frac {2}{3} i K[1]^3-8 i \text {PolyLog}\left (2,i e^{i K[1]}\right ) K[1]+K[1]+8 \text {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 ) \\ \end{align*}