The solution curve of the differential equation yfracdxdy = x(log_e x - log_e y + 1), x > 0, y > 0 passing through the point (e, 1) is

Solution & Explanation

### Core Logic Given DE: fracdxdy = fracxy left(lnleft(fracxyright) + 1right) Let fracxy = t implies x = ty. Differentiating w.r.t y: fracdxdy = t + yfracdtdy ### Step 1: Substitution and Integration t + yfracdtdy = t(ln(t) + 1) = tln t + t yfracdtdy = tln t implies fracdttln t = fracdyy Integrate both sides. Let ln t = p implies frac1t dt = dp. int fracdpp = int fracdyy ln|p| = ln y + C implies ln|ln t| = ln y + C lnleft|lnleft(fracxyright)right| = ln y + C ### Step 2: Applying Boundary Conditions Given curve passes through (e, 1): lnleft|lnleft(frace1right)right| = ln(1) + C implies C = 0 lnleft|lnleft(fracxyright)right| = ln y left|lnleft(fracxyright)right| = e^ln y = y ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Differential Equations

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