Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12

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

#### 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

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