(x+y)dydx=1 dxdy x=y This a linear diffrential equation of the form dxdy+PX=Q IF=e^ ∫ 1dy=e^ y Multiplying by IF xe^ y=∫ye^ ydy+C ...

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Formation of a Differential Equation from a General Solution

Solve: x+ydy...

Question

Solve:
$(x + y) \frac{d y}{d x} = 1$ .

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Solution

$(x + y) \frac{d y}{d x} = 1$
$⟹ \frac{d x}{d y} - x = y$
This a linear diffrential equation of the form
$\frac{d x}{d y} + P X = Q$
$I F = e^{- \int 1 d y} = e^{- y}$
Multiplying by $I F$
$x e^{- y} = \int y e^{- y} d y + C$
$⟹ x e^{- y} = - y e^{- y} - e^{- y} + C$
$⟹ (x + y + 1) e^{- y} = C$
$⟹ x + y + 1 = c e^{y}$

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