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]
\[ \boxed {3 y \cos \left (x \right )+2 x^{3} y+\left (3 \sin \left (x \right )+3\right ) y^{\prime }=-4 x \,{\mathrm e}^{x}} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 395
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)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\frac {\left (-24 x +24 i x \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (-24 i-24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )+4 i x \left (x^{2}+\frac {3}{4}\right ) {\mathrm e}^{i x}-3 x}{3 \,{\mathrm e}^{i x}+3 i}} \left (4 \left (\int \frac {\left (-2 i {\mathrm e}^{\frac {24 x \left ({\mathrm e}^{i x}+i\right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+4 \,{\mathrm e}^{i x} x^{3}+3 i x \,{\mathrm e}^{i x}+24 i \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-3 x -24 \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )}{3 i {\mathrm e}^{i x}-3}}+{\mathrm e}^{\frac {\left (24 i+24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-4 x \left (\left (6 i {\mathrm e}^{i x}-6\right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (i x^{2}-\frac {3}{4}+\frac {3}{4} i\right ) {\mathrm e}^{i x}-\frac {3}{4}-\frac {3 i}{4}\right )}{3 \,{\mathrm e}^{i x}+3 i}}-{\mathrm e}^{\frac {\left (24 i+24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-4 x \left (\left (6 i {\mathrm e}^{i x}-6\right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (i x^{2}-\frac {3}{4}-\frac {3}{4} i\right ) {\mathrm e}^{i x}+\frac {3}{4}-\frac {3 i}{4}\right )}{3 \,{\mathrm e}^{i x}+3 i}}\right ) \left (1-i {\mathrm e}^{i x}\right )^{4 x^{2}} x}{\sin \left (x \right )+1}d x \right )+3 c_{1} \right ) \left (1-i {\mathrm e}^{i x}\right )^{-4 x^{2}}}{3 \left ({\mathrm e}^{i x}+i\right )^{2}} \]
✓ Solution by Mathematica
Time used: 27.47 (sec). Leaf size: 193
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]
\[ y(x)\to \frac {\left (1+i e^{-i x}\right )^{-4 x^2} \exp \left (-8 i x \operatorname {PolyLog}\left (2,-i e^{-i x}\right )-8 \operatorname {PolyLog}\left (3,-i e^{-i x}\right )+\frac {2}{3} x^3 \left (\cot \left (\frac {1}{4} (2 x+\pi )\right )-i\right )\right ) \left (\int _1^x-\frac {4}{3} \exp \left (-\frac {2}{3} \cot \left (\frac {1}{4} (2 K[1]+\pi )\right ) K[1]^3+\frac {2}{3} i K[1]^3+8 i \operatorname {PolyLog}\left (2,-i e^{-i K[1]}\right ) K[1]+K[1]+8 \operatorname {PolyLog}\left (3,-i e^{-i K[1]}\right )\right ) K[1] (i \cos (K[1])+\sin (K[1])+1)^{4 K[1]^2}dK[1]+c_1\right )}{\sin (x)+1} \]