Word Count : 579

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The given differential equation is: [ (7y - 3x + 3),dy + (3y - 7x + 7),dx = 0 ] Rewriting in standard form: [ \frac{dy}{dx} = -\frac{3y - 7x + 7}{7y - 3x + 3} ] This is a linear first-order differential equation in the form of a homogeneous equation with a shift. Let us try the substitution: [ X = x + a, \quad Y = y + b ] so that the equation becomes homogeneous in (X) and (Y). We want constants (a) and (b) such that the constant terms vanish: [ 7Y - 3X + (7b - 3a + 3) = 7Y - 3X ] [ 3Y - 7X + (3b __________ _____ _____ ________ ______ __________ ___ ______.
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