Question

x+y dy/dx=2y solve the differential equation
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Answer 1

Answer:

The solution of the equation x + y(dy/dx) = 2y is:

(A) xy2 = c2 (x + 2y)

(B) y2 = c(x2 + 2y)

(C) log (y – x) = c + [x/(y – x)]

(D) log [x/(x – y)] = c + y – x

Explanation:

x + y(dy/dx) = 2y

∴ (dy/dx) = [(2y – x)/y]

Let y = vx ---- homogeneous differential equation

∴ (dy/dx) = v + x(dy/dx)

∴ v + x ∙ (dv/dx) = [{(2v – 1)x}/{vx}]

∴ x(dv/dx) = [(2v – 1)/v] – v

∴ x(dv/dx) = [(2v – 1 – v2)/v]

∴ [(v ∙ dv)/(v2 – 2v + 1)] = – (dx/x)

∴ ∫ – (dx/x) = ∫+ [(v ∙ dv)/(v2 – 2v + 1)]

∴ ∫ – (dx/x) = (1/2)∫[(2v – 2 + 2)/(v2 – 2v + 1)]dv

∴ ∫ – (dx/x) = (1/2)∫[(2v – 2)/(v2 – 2v + 1)]dv + ∫[(dv)/(v – 1)2]

∴ – log x = (1/2)log (v2 – 2v + 1) – [1/(v – 1)]

∴ [1/(v – 1)] = log x + (log (v – 1)2) × (1/2)

∴ [1/(v – 1)] = log [x(v – 1)]

∴ [x/(y – x)] = log [x{(y/x) – 1}]

∴ [x/(y – x)] = log [y – x] + c

∴ log (y – x)] = c + [x/(y – x)]

Step-by-step explanation:

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Answer 2

Answer:

Given that:

dy/dx=2y/(x-y)

Now , on rearranging you'll get:

dx/dy= (x-y)/2y——————(1) take 'y' common from both numerator & denominator. You’ll get

dx/dy={(x/y)-1}/2

Now put x=vy (say) then x/y=v

Diffrentiate it with respect to y .you'll get

dx/dy=v+y(dv/dy) (according to chain rule: d(uv)/dx=udv/dx+vdu/dx). Now put this value in (1).

v +y(dv/dy)= (v-1)/2 . Multiply both side by 2

2{v+y(dv/dy)}=v-1 or,

2v+2y(dv/dy)=v-1 or,

-2y(dv/dy)=v+1

Now on rearranging the equation we'll get:

dv/(v+1)=-(1/2)dy/y

Now integrate both sides ,

ln(v+1)=(-1/2)lny+lnc , here lnc is a arbitrary comstant.

ln(v+1)= ln{y^(-1/2)×c}

v+1=cy^(-1/2)

Now put the value of v in terms of x and y

(x/y)+1=cy^(-1/2)

x/y= cy^(-1/2)-1

x=y{cy^(-1/2)-1}

x=cy^(1/2)-y