Order and degree — read the highest derivativeकोटि और घात — उच्चतम अवकलज को पढ़िएOrder aur degree — sabse oonche derivative ko padhiye
NCERT 9.2.2 · Class 12 Maths · Chapter 9 · interactive diagram
Parts and what they mean
- Equation (9) of NCERT 9.2.2 (NCERT 9.2.2 का समीकरण (9)) — \frac{d^3y}{dx^3} + 2\left(\frac{d^2y}{dx^2}\right)^2 - \frac{dy}{dx} + y = 0
- Highest order derivative y''' (उच्चतम कोटि अवकलज y''') — The third derivative is the highest one present, so the **order is 3**.
- Order 3, degree 1 (कोटि 3, घात 1) — Order = order of the highest derivative (y'''); degree = its power, which is 1.
- (y'')^2 — not the highest derivative ((y'')^2 — उच्चतम अवकलज नहीं) — The power 2 sits on y'', a lower derivative. The degree looks only at the highest derivative y''', so the degree stays 1.
- Remaining terms (शेष पद) — -y' + y: lower derivatives and y itself; they do not change order or degree.
- Equation (10) of NCERT 9.2.2 (NCERT 9.2.2 का समीकरण (10)) — \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} - \sin^2 y = 0
- (y')^2 — highest derivative, power 2 ((y')^2 — उच्चतम अवकलज, घात 2) — Only y' appears, so the order is 1; its highest power is 2, so the degree is 2.
- +\,y' (+\,y') — Power 1 of y'; the highest power of y' is still 2.
- \sin^2 y — allowed (\sin^2 y — चलेगा) — The equation must be a polynomial in the **derivatives** only. \sin^2 y is a function of y, not of a derivative, so the degree is still defined (NCERT: "not a polynomial in y though").
- Order 1, degree 2 (कोटि 1, घात 2) — Highest derivative y' (order 1) with highest power 2.
- Equation (11) of NCERT 9.2.2 (NCERT 9.2.2 का समीकरण (11)) — \frac{dy}{dx} + \sin\left(\frac{dy}{dx}\right) = 0
- y' and = 0 (y' और = 0) — The only derivative here is the first derivative, so the order is 1.
- \sin(y') — breaks the polynomial (\sin(y') — बहुपद नहीं रहता) — A derivative sits inside sin, so the equation is not a polynomial in y': the degree is **not defined**.
- Order 1, degree not defined (कोटि 1, घात परिभाषित नहीं) — Order is always defined; degree only when the equation is a polynomial in its derivatives.
Original diagram drawn by Simple Learning Hub, based on NCERT 9.2.2. Report a mistake