Page 58 - Standard Handbook Of Petroleum & Natural Gas Engineering

Differential and Integral Calculus 47

Some differential equations of the second order and their solutions follow:

For d2y/dX2 = -n2y
y = C,sin(nx + C,)
= C,sin nx + C,cos nx

For d2y/dx2 = + n2 Y
y = C,sinh(nx + C,)
= C enx + Cqe-"x

For d2y/dx2 = f(y)

where P = jf(y)dy

For d2y/dx2 = f(x)
y = jPdx + C,x + C, where P = jf(x)dx
= XP - jxf(x)dx + C,x + C,

For d2y/dx2 = f(dy/dx), setting dy/dx = z and dP/dx2 = dz/dx
x = jdz/f(z) + C, and
y = jzdz/f(z) + C,, then eliminating z

For d2y/dx2 + Pb(dy/dx) + a2y = 0 (the equation for damped vibration)
If a2 - b2 > 0,

then m = Ja2 - b2
y = C,e-bxsin(mx + C,)
= e-bx[C,sin(mx) + C,cos(mx)]
If a2 - b2 = 0,
y = e-bx(C, + C,x)
If a, - b2 < 0,

then n = db2 - a' and
y = C,e-bxsinh(nx + C )
= C e-(b+n)x + C e-(b-&
For d2y/dx2 + 2b(dy/dx) + a2y = c

y = c/a2 + y,
where y, is the solution of the previous equation with second term zero.
The preceding two equations are examples of linear differential equations with
constant coefficients and their solutions are often found most simply by the use
of Laplace transforms [ 13.
For the linear equation of the nrh order
An(x)d"y/dx" + An-,(x)d"-'y/dxn-' + . . . + A,(x)dy/dx + A,(x)y = E(x)