00:01
Hello students given differential equation is 2x y power 4 e power y plus 2 x y 4 plus y plus x squared y power 4 e power y minus x square y square y square minus 3 minus 3 x multiplied by d y that is equal to 0 so which is of the form which is of the form m d x plus n d y is equal to 0 so where m is equal to m is equal to 2 x y 4 e power y plus 2 x y plus 2 x y plus y plus y plus y plus y n is equal to x square y power 4 e power y minus x square minus x square y square minus 3x so differentiate m with respect to y differentiate partially m with respect to y do m y do y so that is equals to yide x y cube e power y plus 2x y power 4 e power y plus 6 x y square plus 1 next differentiate n with respect to x that is do n by do x is equal to 2 x y power 4 e power y minus 2 x y square minus 3 so here do m by do y is not equal to do n y do x so therefore no x so therefore the given equation given differential equation is not exact since do m y is not equal to do n by do x given differential equation is not exact so the given differential equation is not exact so the given differential equation is not exact so the given differential equation is not exact so we have to make it an exact differential equation.
03:39
So we have 1 divided by n multiplied by do m by do y minus do n -that is equal to 1 divided by x -4 -4 minus.
04:15
X square y square minus 3x multiplied by yid x y cube e power y plus 2x y power 4 e power y plus 6x y square plus 1 minus 2 x y power 4 e power y minus 2 x y square minus 3 so this total is equal to yid x y cube e power y plus yid x y square plus 4 divided by x square y power 4 minus x square y square minus 3x so which is in terms of x and y so this is not equal to f of x not equal to f of x next we have dough n so one divided by m we have 1 divided by m we have 1 divided by m multiplied by do n by do x minus do m by do y is equal to 1 divided by 2 e power 4 e power y plus 2x y cube plus y multiplied by 2x y power 4 e power y minus 2 x y square minus 3 minus of 8 x y cube e power y plus 2 x y power 4 e power y plus 6 x y square plus 1 bracket close so this total is equal to minus 8 x y cube e power y minus 8 x y squared minus 4 divided by 2 x y power 4 e power y plus 2x y cube plus y so in the numerator take minus 4 here 2x 2x y cube e power y minus plus 2x y squared plus 2x y squared plus 1 divided by in the denominator y is common 2x y cube e power y plus 2x y square plus 1 that is equal to minus 4 by y so, this is in terms of y, so that is equal to f of y.
08:52
So therefore, integrating factor is equal to integrating factor is equal to, is equal to, is equal to e power integral of f of y, d, y.
09:18
So that is equal to e power integral of f of y is minus 4 divided by y, d y, that is equal to e power integral.
09:29
Minus 4 of integral of 1 divided by y and y.
09:34
So that is equals to e power minus 4 log y.
09:41
So that is e power log y power minus 4 that is e power log 1 by 4.
09:52
So integrating factor is equal to 1 by y power 4.
09:59
So here y power 4 integrating factor is 1 divided by y power 4 multiplied the given differential equation by multiply the given differential equation by 1 divided by y by 4 so the given differential equation 2 x y power 4 e power y power y plus 2 2xy cube plus y divided by y power y power 4 multiplied by d x plus x squared y power 4 e power y minus x squared y squared minus 3x divided by y power 4 multiplied by d y is equal to 0 so which is of the form m1 d x plus n1 dy is equal to 0 so which is of the form m1 d x plus n1 dy is equal to 0 so which is of the form m1 d x plus n1 equal to 0.
11:42
So, differentiate m with respect, m with respect to y...