Differential Equations Question - None

Question

Find the solution of the differential equation \frac{dy}{dx} = \frac{x + y + 3}{2x + 2y + 1}

Accepted Answer

Step 1: Note that \frac{a}{a_1} = \frac{b}{b_1} = \frac{1}{2}, meaning the coefficients of x and y in the numerator and denominator are proportional.

Step 2: Use the substitution v = x + y. Then 1 + \frac{dy}{dx} = \frac{dv}{dx}.

Step 3: The given equation becomes:

\frac{dv}{dx} - 1 = \frac{v + 3}{2v + 1} or \frac{dv}{dx} = \frac{v + 3}{2v + 1} + 1 = \frac{3v + 4}{2v + 1}

Step 4: Separate variables:

\frac{2v + 1}{3v + 4}dv = dx

Step 5: Integrate both sides. Using partial fractions, \frac{2v + 1}{3v + 4} = \frac{2}{3} - \frac{5}{3(3v + 4)}

Step 6: On integrating:

x + C = \frac{2}{3}v - \frac{5}{9}\log(3v + 4) = \frac{2}{3}(x + y) - \frac{5}{9}\log(3x + 3y + 4)

∴ Answer is D.

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