(H + Kx) cos x + (M + Nx) sin x,
where H, K, M, N are arbitrary constants. To obtain a particular integral we must find a value of (1 + D²)−²x³ cos x; this is the real part of (1 + D²)−² eixx³ and hence of eix [1 + (D + i)²]−² x³
or
eix [2iD(1 + ½iD)]−² x³,
or
−¼eix D−² (1 + iD − ¾D² − ½iD³ + 5⁄16D4 + 3⁄16iD5 ...)x³,
or
−¼eix (1⁄20x5 + ¼ix4 − ¾x³ − 3⁄2 ix² + 15⁄8 x + 9⁄8 i);
the real part of this is
−¼ (1⁄20 x5 − ¾x² + 15⁄8x) cos x + ¼ (¼x4 − 3⁄4x² + 9⁄8) sin x.