Solveeit Logo
Login

Question

Question: The solution of \(\frac{dy}{dx} = \frac{x - 3y + 2}{3x - y + 6}\) is...

The solution of dydx=x−3y+23x−y+6\frac{dy}{dx} = \frac{x - 3y + 2}{3x - y + 6} is

A

y2+6(x+2)y+(x+2)2=cy^{2} + 6(x + 2)y + (x + 2)^{2} = c

B

y2−6(x+2)y+(x+2)2=cy^{2} - 6(x + 2)y + (x + 2)^{2} = c

C

y2−6(y+2)x+x2=cy^{2} - 6(y + 2)x + x^{2} = c

D

None of these

Answer

y2−6(x+2)y+(x+2)2=cy^{2} - 6(x + 2)y + (x + 2)^{2} = c

Explanation

Solution

Given equation is non-homogeneous

Let x = X + h, y = Y + k

⇒ dydx=dYdX\frac{dy}{dx} = \frac{dY}{dX}

∴ dYdX=(X+h)−3(Y+k)+23(X+h)−(Y+k)+6=X−3Y+(h−3k+2)3X−Y+(3h−k+6)\frac{dY}{dX} = \frac{(X + h) - 3(Y + k) + 2}{3(X + h) - (Y + k) + 6} = \frac{X - 3Y + (h - 3k + 2)}{3X - Y + (3h - k + 6)}

Let us select h and k so that h – 3k + 2 = 0 and

3h – k + 6 = 0

Solving, k = 0, h = – 2 ∴ X = x – h = x + 2, Y=y−k=yY = y - k = y

∴ dYdX=X−3Y3X−Y\frac{dY}{dX} = \frac{X - 3Y}{3X - Y}, which is homogeneous

Now, let Y = vX

⇒ dYdX=v+XdvdX\frac{dY}{dX} = v + X\frac{dv}{dX} ⇒ X−3Y3X−Y=v+XdvdX\frac{X - 3Y}{3X - Y} = v + X\frac{dv}{dX}

⇒ 1−3(Y/X)3−(Y/X)=v+XdvdX\frac{1 - 3(Y/X)}{3 - (Y/X)} = v + X\frac{dv}{dX} ⇒ 1−3v3−v=v+XdvdX\frac{1 - 3v}{3 - v} = v + X\frac{dv}{dX}

⇒ XdvdX=1−3v3−v−v=v2−6v+13−vX\frac{dv}{dX} = \frac{1 - 3v}{3 - v} - v = \frac{v^{2} - 6v + 1}{3 - v} ⇒ (3−v)dvv2−6v+1=dXX\frac{(3 - v)dv}{v^{2} - 6v + 1} = \frac{dX}{X}

⇒ 2v−6v2−6v+1dv=−2dXX\frac{2v - 6}{v^{2} - 6v + 1}dv = - 2\frac{dX}{X}

Integrating, ln⁡(v2−6v+1)=−2ln⁡X+ln⁡c\ln(v^{2} - 6v + 1) = - 2\ln X + \ln c

⇒ ln⁡(v2−6v+1)+ln⁡X2=ln⁡c\ln(v^{2} - 6v + 1) + \ln X^{2} = \ln c ⇒ X2(v2−6v+1)=cX^{2}(v^{2} - 6v + 1) = c

⇒ Y2−6XY+X2=cY^{2} - 6XY + X^{2} = c

∴ y2−6(x+2)y+(x+2)2=cy^{2} - 6(x + 2)y + (x + 2)^{2} = c