Painevg

2022-01-19

Find the general solution of the differential equation
${y}^{\prime }=\frac{x-y+3}{x-y}$

Terry Ray

Expert

A differential equation is an equation in x,y and derivatives of y. A first order differential equation is on in which the highest order derivative present in the differential equation is first order.
A differential equation in some cases can be simplified to a simpler differential equation with an appropriate substitution. For given differential equation find an appropriate substitution to simplify the differential equation.
Given differential equation is ${y}^{\prime }=\frac{x-y+3}{x-y}$. Use the substitution x-y=u. Differentiating this gives $1-\frac{dy}{dx}=\frac{du}{dx}$. This can be rewritten as $1-\frac{du}{dx}=\frac{dy}{dx}$. Substitute this in the differential equation and integrate.
$1-\frac{du}{dx}=\frac{u+3}{u}$
$1-\frac{du}{dx}=1+\frac{3}{u}$
$-\frac{du}{dx}=\frac{3}{u}$
$-udu=3dx$
$\frac{{u}^{2}}{2}=3x+C$
${u}^{2}=-6x-2C$
$u=\sqrt{-6x-2C}$
$x-y=\sqrt{-6x-2C}$
$y=x-\sqrt{-6x-2C}$
Hence, general solution to the differential equation is $y=x-\sqrt{-6x-2C}$

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