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]
\[ \boxed {3 \cos \left (x \right ) y+4 \,{\mathrm e}^{x} x +2 x^{3} y+\left (3 \sin \left (x \right )+3\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 488
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)
\[ y \left (x \right ) = \frac {\left (1-i {\mathrm e}^{i x}\right )^{-4 x^{2}} \left (\int -\frac {4 x \left (1-i {\mathrm e}^{i x}\right )^{4 x^{2}} \left ({\mathrm e}^{-\frac {4 i x^{3} {\mathrm e}^{i x}+24 i x \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}+3 x -3 i x \,{\mathrm e}^{i x}-24 i \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-3 i x -24 \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-3 x \,{\mathrm e}^{i x}-24 x \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )}{3 \left ({\mathrm e}^{i x}+i\right )}}+2 i {\mathrm e}^{-\frac {4 i x^{3} {\mathrm e}^{i x}+24 i x \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 i \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-3 i x -24 \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-3 x \,{\mathrm e}^{i x}-24 x \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )}{3 \left ({\mathrm e}^{i x}+i\right )}}-{\mathrm e}^{-\frac {4 i x^{3} {\mathrm e}^{i x}+24 i x \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 x \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+3 i x \,{\mathrm e}^{i x}-3 x -24 \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 i \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-3 x \,{\mathrm e}^{i x}-3 i x}{3 \left ({\mathrm e}^{i x}+i\right )}}\right )}{3 \sin \left (x \right )+3}d x +c_{1} \right ) {\mathrm e}^{\frac {4 i x^{3} {\mathrm e}^{i x}+24 i x \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 x \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+3 i x \,{\mathrm e}^{i x}-3 x -24 \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-24 i \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )}{3 \,{\mathrm e}^{i x}+3 i}}}{\left ({\mathrm e}^{i x}+i\right )^{2}} \]
✓ Solution by Mathematica
Time used: 26.035 (sec). Leaf size: 182
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 \frac {\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 {4 x^3}{3 e^{i x}+3 i}\right ) (\sin (x)+i \cos (x)+1)^{-4 x^2} \left (\int _1^x-\frac {4}{3} \exp \left (-\frac {4 K[1]^3}{3 i+3 e^{i K[1]}}+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} \\ \end{align*}