Question

solve the differential equation D²-a²=cosh ax​

Answer 1

Step-by-step explanation:

The given equation in its symbolic form is D2 + a2 y = cosec ax Thus the auxiliary equation becomes

D2 + a2 = 0 i.e. D = ±ia∴ C.F. = c1cos ax + c2sin axFor P.I = 1/D2 + a2 cosec ax= 1/D2 --a2 cos ec ax= 1/D + iaD - iacosec ax= 1/2ia1/D -ia - 1/D + ia cosec axBy partial FractionsOn the same lines changing i to –i

we get1/D + ia cosec ax = e-iax[logsin ax/a + ix]Using 1 and 2

Hence the complete solution isy = c1 cos ax + c2 sin ax + 1/a logsin ax/a sin ax - x cos ax