00:01
To solve the given differential equations, we need to find an integrating factor that makes the left -hand side of the equation and x are differential.
00:07
The integrating factor is given by mu of x is equal to e power integral p of x into dx.
00:18
So, p of x is the coefficient of dx and q of x is the coefficient of dy in the differential equation.
00:23
So here, p of x is equal to x plus y minus 1 and q of x is equal to 2x plus 2y plus 1.
00:34
The integrating factor is mu of x is equal to e power integral p of x dx.
00:41
So, that is equal to e power integral of x plus y minus 1 into dx, which is equal to e power x square by 2 plus xy minus x.
00:53
Multiplying both sides of the differential equation by the integrating factor, we get e power x square by 2 plus xy minus x into x plus y minus 1 into dx plus e power x square by 2 plus xy minus 2 into 2x plus 2y plus 1 into dy, which is equal to 0.
01:28
This can be written as p of e power x square by 2 plus xy minus x into x plus y is equal to 0.
01:39
So, integrating both sides, we get e power x square by 2 plus xy minus x of x plus y is equal to c.
01:52
C is the constant of the integration.
01:54
Simplifying this expression, we can write e power x square by 2 plus xy minus x of x plus y is equal to c.
02:04
This is the answer for 1.
02:06
So, in second question, 6x minus 3y plus 2 dx plus 3 into x plus 3y minus 4 dy.
02:12
So, p of x is 6x minus 3y plus 2.
02:14
Q of x is 3 into x plus 3y minus 4.
02:17
Then mu of x is equal to e power integral, that is, e power 3x square minus 3xy plus 2x.
02:26
Then multiplying both sides of the differential equation by the differentiating factor, we get e power 3x square minus 3xy plus 2x into 6x minus 3y plus 2 into dx plus e power 3x square minus 3xy plus 2x into 3 into x plus 3y minus 4 into dy is equal to 0...