Example 7: the family y = x² + log|x| + C and the curve through (1, 1)उदाहरण 7: कुल y = x² + log|x| + C और (1, 1) से जाने वाला वक्रExample 7: family y = x² + log|x| + C aur (1, 1) se jaane wala curve

NCERT Example 7 · Class 12 Maths · Chapter 9 · interactive diagram

Parts and what they mean

• Differential equation of Example 7 (उदाहरण 7 का अवकल समीकरण) — x\,dy = (2x^2 + 1)\,dx, x \ne 0, i.e. \frac{dy}{dx} = 2x + \frac{1}{x}.
• Family y = x^2 + \log|x| + C (कुल y = x^2 + \log|x| + C) — Changing C slides the same curve up or down: at a given x every member has the same slope 2x + \frac1x.
• Required curve y = x^2 + \log|x| (अभीष्ट वक्र y = x^2 + \log|x|) — Putting x = 1, y = 1: 1 = 1 + 0 + C, so C = 0.
• Point (1, 1) (बिंदु (1, 1)) — The curve must pass through it; this fixes C.
• x = 0 is not allowed (x = 0 वर्जित है) — \log|x| is not defined at x = 0; each curve has two separate branches, for x < 0 and x > 0.
• x-axis (x-अक्ष) — The horizontal number line; inputs x are read here.
• y-axis (y-अक्ष) — The vertical number line; outputs y = f(x) are read here.
• Origin O(0, 0) (मूल बिंदु O(0, 0)) — Where the two axes cross.

Original diagram drawn by Simple Learning Hub, based on NCERT Example 7. Report a mistake