Question

2. Solve the initial value problem: (2.1 – y) + (2y – r)y' = 0) with y(1) = 3. 3. Find the numerical value of b that makes the following differential equation exact. Then solve the differential equation using that value of b. (xy? + br’y) + (x + y)x+y = 0

exact differential equations

Answer 1

2. The given initial value problem is (24-7) + (24-u) y'=0 with y(1) = 3 (2n-1)+(2ywy'=0 (24-y) + (24) dy (2n-y) du+ (29-n) dyzo comparing with Mau+Ndy=0 M=22-4 N=2y-n am og ON an = Here am sy = IN The given differential equation is exact. Now, Suda S (207) du y= constant y=constant x-ny s(terms of N not containing x) dy = S2y dy I I .. The general solution is I t I c .. (43 ny) + y² = 0 given, y(1)=3 :. (13- (1) (3)) +(37=0 (= 7 putting this value of Cin , we get (x²-ny+y2=

--0 3. (my"+ 6x²y) + [n+y) m² y = 0 → (uy+bu?y) +(u?+u?y) december → (My² + bx²y du+ (x²+ hydy=o comparing with Mdu tNay=0 Many ²+bx²y Naulthy OM SN = 3x²+2xy The given differential equation is exact, if JM ay any+bu? =ON du zny+bu?=3u²+zny br²=4² b=3 For b=3, :1 May Many2 +3ity, N=42+x² scny² + 3y) an y=constant y = constant '+ x3y Sterus of a not containing a) ay = Jody I the general solution is xy + x²y = 6 x²y2 + 2x3y=(,, where (=20