Schooling · Class 12 · Mathematics · Differential Equations · Chapter on a page
CBSE Class 12 · Mathematics · NCERT Mathematics Part-II
Differential Equations: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- A differential equation contains…
- order
- The number of arbitrary constants in…
- Variable-separable
- homogeneous
- linear first-order
- particular solution
- Formulas and facts
- Order
- Degree
- Variable-separable
- Homogeneous substitution
- Linear equation form
- Integrating factor (IF)
- Solution of linear equation
- Worked examples
- Finding order and degree
- Solving a variable-separable equation
- Solving a linear first-order equation
- Watch out for
- Forgetting to remove a square root…
- Dropping the arbitrary constant…
- Substituting y = vx but leaving…
- Using IF = ∫P dx instead of IF =…
- Mixing up order and degree
What this chapter is about
A differential equation is an equation that involves an unknown function and one or more of its derivatives. In Class 12, you learn to identify, form and solve such equations. This chapter connects your knowledge of differentiation and integration: differentiation creates the equation from a family of curves, and integration recovers the solution.
Key ideas
- A differential equation contains derivatives such as dy/dx, d²y/dx², etc., along with x, y and constants.
- The order is the highest derivative that appears; the degree is the power of that highest derivative after the equation is made free of radicals and fractions in derivatives.
- The number of arbitrary constants in the general solution equals the order of the equation.
- Variable-separable equations can be written as f(y) dy = g(x) dx; integrate both sides directly.
- A homogeneous equation of degree zero in x and y is solved by the substitution y = vx (or x = vy), which converts it to variable-separable form.
- A linear first-order equation dy/dx + P(x)y = Q(x) is solved using the integrating factor IF = e^(∫P dx); multiply through and integrate.
- A particular solution is obtained by assigning specific values to the arbitrary constants using initial or boundary conditions.
Formulas and facts to remember
- Item: Order · Formula / Rule: Highest derivative in the equation · Meaning: Tells how many integrations are needed in principle
- Item: Degree · Formula / Rule: Exponent of highest-order derivative (equation polynomial in derivatives) · Meaning: Must remove radicals/fractions first
- Item: Variable-separable · Formula / Rule: ∫f(y) dy = ∫g(x) dx + C · Meaning: Separate y-terms to one side, x-terms to the other
- Item: Homogeneous substitution · Formula / Rule: y = vx ⇒ dy/dx = v + x dv/dx · Meaning: Converts dy/dx = F(y/x) to separable form
- Item: Linear equation form · Formula / Rule: dy/dx + Py = Q · Meaning: P and Q are functions of x only
- Item: Integrating factor (IF) · Formula / Rule: IF = e^(∫P dx) · Meaning: Multiply entire equation by IF
- Item: Solution of linear equation · Formula / Rule: y × IF = ∫(Q × IF) dx + C · Meaning: After multiplying by IF, left side is d/dx (y × IF)
Common mistakes
- Forgetting to remove a square root before finding the degree → clear radicals first so the equation is polynomial in derivatives.
- Dropping the arbitrary constant after integration → always include + C; its value is fixed only when an initial condition is given.
- Substituting y = vx but leaving dy/dx unchanged → replace dy/dx by v + x dv/dx.
- Using IF = ∫P dx instead of IF = e^(∫P dx) → the integrating factor is an exponential, not the integral itself.
- Mixing up order and degree → order counts which derivative, degree counts its power.
Quick revision
- Order = highest derivative present; Degree = power of that derivative (after rationalising).
- Variable-separable: move all y and dy to one side, all x and dx to the other, then integrate.
- Homogeneous: substitute y = vx, simplify, separate, integrate, replace v by y/x.
- Linear dy/dx + Py = Q: multiply by IF = e^(∫P dx), then integrate to get y × IF = ∫Q × IF dx + C.
- Number of arbitrary constants in general solution = order of the equation.
- Apply initial conditions only after finding the general solution to obtain the particular solution.