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Definition:
If M and N are functions of x and y, the equation M dx+ N dy=0….(1)
is called exact when there exists a function f(x,y) of x
and y, such that
d[f(x,y)] = M dx + N dy,. …..(2)
WORKING RULE:
Compare the given equation with
M dx N dy =0 and find M and N. Then find out dM/dy and dM/dx, if dM/dy=dN/dx, we conclude that the given equation is exact. If the equation is exact, then
Step. 1) Integers M with respect to x treating y as a constant.
Step. 2) Integrate with respect to y only those terms of N which do not contain x.
Step. 3) Equate the sum of these two integrals (found steps 1 and 2) to an arbitrary constant and thus we obtain the required solution. In short the solution of Exact equation
M dx+ N dy =0 is
∫ M dx + ∫ (in terms in N not containing x) dy = c. where c is sn arbitrary constant.
Example. 1 :
1)(x²-4xy-2y²)dx + (y²-4xy-2x²)dy=0
here M= x²-4xy-2y² and N=y²-4xy-2x²
dM/dy= -4x-4y and dN/dx= -4y-4x
so that dM/dy= dN/dx
hence, the given equation is exact hence its solution is.
∫ M dx + ∫(terms in N not containing x) dy = c′
∫(x² -4xy -2y²)dx + ∫y² dy =c′
x³/3 -4y(x²/2)-2y²x + y³/3 = c/3
taking c′ = c/3
Example. 2)
(2x-y+1)dx + (2y-x-1) dy= 0
comparing Mdx +N dy=0
M=2x-y+1 N= 2y-x-1
dM/dy= -1= dN/dx hence exact
∫M dx(treating y as constant) +∫(terms in N not containing x)dy =c
∫(2x-y+1)dx + ∫(2y-1)dy=0
or x² - xy+x -y² -y =c
EXERCISE
1)(4x+3y+1) dx +(3x+2y+1) dy =0
2) dy/dx = (2x-y+1)/(x+2y-3)
3) (1+eˣ/ʸ)dx + eˣ/ʸ {1-(x/y)} dy = 0
4) ₍ᵥ2ₑxy² ₊₄ₓ³₎dx ₊ ₍₂ₓᵥₑxy² ₋ ₃ᵥ²₎dy ₌ ₀
5) (ax +by+g) dx +(hx+ by +f)dy =0
6) {y(1+ 1/x) + cos y}dx + {x+ log x -x siny) dy = 0
7) xdx+ydy +(x dy - y dx)/(x² + y²)= 0
8) x dx + y dy= a²(x dy -y dx)/(x² +y²)
9) (x dy + ydy)(x² +y²) = a²(x dy - ydx)
10) (x³+xy²+a²y)dx +(y³+yx²-a²x)dy=0
11) (x²+y²)(xdx+ydy)=xdy - ydx.
12) (r+sinθ - cosθ)dr + r(sinθ+cosθ)dθ=0
13) y sin 2x dx -(1+y²+cos²x) dy=0
14) y sin 2x dx - (y² + cos²x) dy = 0
15) (x² +y² +x)dx - ((2x² +2y² -y)dy=0
16) (4x +3y +1) dx +(3x+2y+1) dy=0
17) Find the value of constant K such that (2xeʸ +3y²)(dy/dx)+(3x²+ Keˣ)=0 is a exact . further, for this of K solve the equation.
18) (x+2y -2)dx + (2x-y+3) dy=0
19) (2ax+by)ydx + (ax+2by)x dy=0
20) (x²- ay)dx = (ax -y²) dy
21) dy/dx = (2x -y)/(x+2y - 5)
22) (x² +y² +a²)y dy +(x²+y²- a²)dx=0
23) (eˣ +1) cos x dx+ eʸsinx dy =0
24) x(x² + 3y²)dx + y(y²+3x²)dy =0
25) (a² - 2xy -y²) dx - (x+y)² dy =0
26) (3x² +6xy²) dx + (6x²y²+ 4y³)dy=0
27) (x⁴-2xy²+y⁴)dx - (2x²y -4xy² +siny) dy =0
28) (3x² + 4xy)dx + (2x² + 2y) dy =0
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