Exact Differential Equations

An exact differential equation is formed by directly differentiating its primitive (solution) without any other process.
Mdx + Ndy = 0
is said to be an exact differential equation if it satisfies the following condition
∂M/∂y = ∂N/∂x
where ∂M/∂y denotes the differential coefficient of M with respect to y keeping x constant and ∂N/∂x is the differential coefficient of N with respect to x keeping y constant.

Method for solving Exact Differential Equations
  1. Integrate M w.r.t. x keeping y constant.
  2. Integrate w.r.t. y only those terms of N which do not contain x.
  3. Result of I + Result of II = Constant.

Equations reducible to exact equations

Sometimes a differential equation Mdx + Ndy = 0 which is not exact may become so, on multiplication by a suitable function known as the integrating factor.

  • if [∂M/∂y - ∂N/∂x]/N is a function of x alone, say f(x), then I.F. = e∫f(x)dx

  • if [∂N/∂x - ∂M/∂y]/M is a function of y alone, say f(y), then I.F. = e∫f(y)dy

  • if M is of the form M = yf1(xy) and N is of the form N = xf2(xy) then I.F. = 1/(Mx-Ny)

  • For this type of xmyn(aydx + bxdy) + xm'yn'(a'ydx + b'xdy) = 0, the integrating factor is xhyk.
    I.F. = xh.yk
    where (m + h + 1)/a = (n + k + 1)/b and (m' + h + 1)/a' = (n' + k + 1)/b'

  • Homogeneous equation
    If Mdx + Ndy = 0 be a homogeneous equation in x and y, then
    I.F. = 1/(Mx + Ny)