Slope field of dy/dx = x + xy and the curve through (0, 1)dy/dx = x + xy का ढाल-क्षेत्र और (0, 1) से जाने वाला वक्रdy/dx = x + xy ka slope field aur (0, 1) se jaane wala curve

NCERT Example 18 · Class 12 Maths · Chapter 9 · interactive diagram

Parts and what they mean

• Example 18: \frac{dy}{dx} = x + xy (उदाहरण 18: \frac{dy}{dx} = x + xy) — The equation tells the slope of the tangent at every point (x, y).
• Slope field (little tangent dashes) (ढाल-क्षेत्र (छोटी स्पर्श-रेखाएँ)) — At each grid point we draw a tiny line with slope x + xy. A solution curve must follow these dashes everywhere.
• Particular solution y = -1 + 2e^{x^2/2} (विशिष्ट हल y = -1 + 2e^{x^2/2}) — General solution y = -1 + Ce^{x^2/2}; through (0, 1): 1 = -1 + C, so C = 2 (NCERT Example 18).
• Another member: y = -1 - e^{x^2/2} (एक और सदस्य: y = -1 - e^{x^2/2}) — The curve of the same family with C = -1; it passes through (0, -2).
• C = 0: the line y = -1 (C = 0: रेखा y = -1) — Here x + xy = x(1 + y) = 0, so the dashes are flat along it and it is itself a solution.
• Point (0, 1) (बिंदु (0, 1)) — Start here and follow the dashes: you trace the orange curve.
• 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 18. Report a mistake