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Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations

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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

Shaalaa.com | Differential Equation part 17 (1st order linear differential Equation)

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Differential Equation part 17 (1st order linear differential Equation) [00:11:43]
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