Question
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Answer :
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 - 7a + 7) = 3Y - 7X ] Solving for (a) and (b): 1. (7b - 3a + 3 = 0 \implies 7b - 3a = -3) 2. (3b - 7a + 7 = 0 \implies 3b - 7a = -7) Multiply first equation by 3 and second by 7 to eliminate (b): [ 21b - 9a = -9 ] [ 21b - 49a = -49 ] Subtract the first from the second: [ -40a = -40 \implies a = 1 ] [ 7b - 3(1) =
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