Question

1B-1. Test the following ODE's for exactness, and find the general solution for those which are exact. a) 3x2y dx + (23 +) dy = 0 b) (22 - y2) dx + (y2 – 22) dy = 0 c) veuvdu + yew dv = 0 d) 2xy dx – 22 dy = 0

Answer 1

O Giren ODES: @ → 3xy dx + (x3 +43) dy=0 compared this ODE with Max+Ndy = 0, we get M = 3ohy, N= x²443 1 for be The necessary & Bufficient condition the ODE Mdx+ Ndy=0 to exact is I ax 3 - ODE . Now, DM = 3 ; = 3x2 - M = 2N for the above so the ODE (a) is exact. The general solution is Mdx + ((Terms not containging) day 1 x în N (Taking constant) y (3 x) dx + S (43) dy = 0 [x3y + 1 / 4 4 4 = constant ! y as (1= constant)

(6) (2² 4²) dx + (29²x²) dy=0 - Comparing the ODE with the ODE Max+ Ndy o we get M = x²y² , N=y= 22 e Now , OM = -24, 2N = -2x So IM + 2N for the ODE (6), which means the ODE (b) is not exact in its given form. Now, if xfy, then we can write (b, by dividing ~xty2 - dy - dx = 0 Now M=1, N=-1 an Jy - Oz AN the - dy-dx=0 is exact general solution is y-x= constant

(do 2xy dx-vdy=0 comparing ODE (d) with Max+ Ndy=0 we get M = 284, N2-22 IM IN Now , AM = 2x, ax = -2x so the d. ODE & is not exact in its gireen form. However we can find an integrating factor (which can make the given ODE by the formula below: exact)

1 * If Mxt Ny 70 & the ODE be homogeneous, then I ntny) is an integrating factor of the canalion Max+ Ndy =or Here, the giveen ODE 2xy dx-rady=0 is homogeneous (as M=28% , N=- both are of degree 2) & Mx + Ny = 2 y = xy to.. in Integrating factor (I.F.) = y. multiplying both sides of the above ODE by - we get 2 dx - 1 dy =0 which is exact. Integrating :- 2 ln(x) - ln (as) = lnce), In(e) = arbitrary constant. -cl, which is general solution. the required

a Gireen - - (3 + x) dy + y dx = 0 - ① comparing ② with Maxt Ndy = 0, we get M=Y, Now, N=-(3 + x) , - So Ran is not exact in its current form. we have to find an integrating factor which can makes exact. [ (*) If th ( 32 - 2 ) = $("), ie, a function of y alone, then So (2) dy is an integrating factor of the ODE.

Here, 1/2 M ax -- y - y alone. - 11/23 (-1-1) -- 2 = function of the integrating factor is: I Fi = eS ( 2 ) dy - 2:28nco) multiplying ③ by y, we get 13 dx - (3 + 2/2) dy = 0 dre - zdy - gdy=0 d (3 - 1 2 4 2 ) = 0 , which is exact. Integrating :- we get - Žy? constant which is the general solution of the ODE (given).