#### notes

A differential equation of the from `(dy)/(dx) + Py = Q`

To solve the first order linear differential equation of the type

`(dy)/(dx) + Py= Q` ...(1)

Multiply both sides of the equation by a function of x say g (x) to get

g(x)`(dy)/(dx) + P.(g(x)) y = Q . g(x)` ...(2)

Choose g(x) in such a way that R.H.S. becomes a derivative of y . g (x).

i.e. `g(x)(dy)/(dx) + P.g(x)y = d/(dx) [y.g(x)]`

or `g(x) (dy)/(dx) + P.g(x)y ` = g(x)`(dy)/(dx) + y g'(x)`

`=> P.g(x) = g'(x)`

o r P = g'(x)/g(x)

Integrating both sides with respect to x, we get

`int Pdx = int (g'(x))/g(x)dx`

or `int P.dx = log (g(x))`

or g(x) = `e^(int Pdx)`

On multiplying the equation (1) by g(x) =`e^( int Pdx)` , the L.H.S. becomes the derivative of some function of x and y. This function

g(x) = `e^(int P dx)` is called Interrating Factor (I.F.) of the given differential equation.

Substituting the value of g (x) in equation (2), we get

`e^(Pdx) (dy)/(dx) + Pe^(int Pdx) y = Q . e^(Pdx)`

Or `d/(dx) (ye^(intPdx)) = Qe^(int Pdx)`

Integrating both sides with respect to x, we get

`y.e^(int P dx) = int (Q.e^(int P dx)) dx` Or

`y = e^(-int Pdx) = int (Q.e^(int P dx)) dx + C`

which is the general solution of the differential equation.

**Steps involved to solve first order linear differential equation:**

(i) Write the given differential equation in the form `(dy)/(dx)` + Py = Q where P, Q are constants or functions of x only.

(ii) Find the Integrating Factor (I.F) = `e^(int Pdx)`

(iii) Write the solution of the given differential equation as

y (I.F) = `int`(Q × I.F )dx + C

In case, the first order linear differential equation is in the form `(dx)/(dy) + P_1x = Q_1`, where, `P_1` and `Q_1` are constants or functions of y only.

Then I.F = `e^(P_idy)` and the solution of the differential equation is given by

x . (I.F) = `int (Q_1 xx I.F)` dy + C