Chapter 9 — Differential Equations

Chapter 9 of the Class 12 Maths NCERT textbook, "Differential Equations", teaches how to form and solve first-order differential equations using three methods: variables separable, homogeneous, and linear differential equations.

• Describing equations with derivatives — A differential equation ties a function to its own rates of change. The chapter classifies such equations by order (the highest derivative) and degree (its highest power), giving students a vocabulary for recognising what kind of equation they face.
• General versus particular solutions — Solving a differential equation yields a whole family of curves carrying arbitrary constants — the general solution. Fixing those constants with given conditions singles out one particular solution, mirroring how integration always leaves a constant to be pinned down.
• Three solving strategies — The chapter matches each equation type to a method: separating variables when they split cleanly, the substitution y = vx for homogeneous equations, and the integrating-factor method for linear equations — with applications to population growth, compound interest and curve geometry.

1. 01Order of a differential equation is defined as the order of the highest order derivative of the dependent variable present in the equation; order and degree are always positive integers when defined.
2. 02Degree is the highest power of the highest order derivative, but only when the equation is a polynomial in its derivatives; equations like y′′′ + y² + e^(y′) = 0 have undefined degree.
3. 03The general solution contains as many arbitrary constants as the order of the equation; a particular solution is obtained by assigning specific values to those constants.
4. 04Variables separable method rewrites dy/dx = h(y)·g(x) as (1/h(y))dy = g(x)dx and integrates both sides independently.
5. 05A homogeneous differential equation has F(x,y) as a homogeneous function of degree zero; it is solved by substituting y = vx (or x = vy), which reduces it to a separable equation.
6. 06A first-order linear differential equation dy/dx + Py = Q is solved by multiplying through by the integrating factor e^(∫P dx) and integrating both sides.