# Differential Equations Notes, Class 12 Mathematics | SwaVid Differential Equations, NCERT Class 12 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers. Canonical: https://www.swavid.com/cbse/class-12/maths/chapter/differential-equations Source: https://www.swavid.com/cbse/class-12/maths/chapter/differential-equations # Differential Equations ## Formulas and definitions ## Key ideas ## Common mistakes ## Quick checks ## Keep going ## Questions parents ask ## Class 12 Mathematics on SwaVid ### Basic concepts and definitions ### Order and degree ### General and particular solutions ### Variable separable method ### Homogeneous differential equations ### Linear differential equations ### Next chapter ### Related chapter ### Related chapter ### Related chapter ### All chapters SwaVid › CBSE › Class 12 › Mathematics › Differential Equations Notes written by SwaVid from the NCERT textbook. Class 12 Mathematics, Chapter 9. Key ideas, formulas and common mistakes. CBSE does not print marks for a single chapter. Short answer. This chapter introduces differential equations along with their order, degree, and solutions. Students learn to find general and particular solutions using variable separation, homogeneous equations, and first order linear differential equations. Source: NCERT Mathematics Part II (Reprint 2026-27) , Chapter 9, Differential Equations An equation involving an independent variable, dependent variable, and derivatives of the dependent variable with respect to the independent variable is called a differential equation. An ordinary differential equation involves derivatives with respect to only one independent variable. The order of a differential equation is the order of the highest order derivative occurring in it. The degree is the highest power of the highest order derivative when the differential equation is a polynomial equation in its derivatives. A general solution contains as many arbitrary constants as the order of the differential equation. A particular solution is obtained from the general solution by assigning specific values to these arbitrary constants based on given conditions. A first order differential equation of the form dy / dx = f(x) g(y) is solved by separating variables into h(y) dy = f(x) dx, where h(y) = 1 / g(y). Both sides are then integrated directly. A differential equation dy / dx = F(x, y) is homogeneous if F(x, y) is a homogeneous function of degree zero. It is solved by substituting y = v x, which transforms the equation into a variable separable form in v and x. A first order linear differential equation has the form dy / dx + P y = Q, where P and Q are constants or functions of x only. Multiplying by the integrating factor e^(∫ P dx) enables direct integration. Try each question first. Then tap it to see the answer. Identify the highest derivative: d²y / dx², which has order 2. Check if the equation is a polynomial in its derivatives: the term cos(dy / dx) prevents it from being a polynomial in derivatives. Therefore, the order is 2 and the degree is not defined. Order = 2, Degree is not defined Separate the variables: dy / (1 + y²) = dx / (1 + x²). Integrate both sides: ∫ dy / (1 + y²) = ∫ dx / (1 + x²). Evaluate the integrals: tan⁻¹(y) = tan⁻¹(x) + C. tan⁻¹(y) = tan⁻¹(x) + C Compare with dy / dx + P y = Q: here P = 1 / x and Q = x². Compute ∫ P dx = ∫ (1 / x) dx = log|x|. Calculate the integrating factor: I.F. = e^(∫ P dx) = e^(log|x|) = x (for x > 0). x Vector Algebra . Continuity and Differentiability . Integrals . Application of Integrals . The 13 chapters of Class 12 Mathematics . CBSE does not give marks to one chapter. It prints marks for units or parts, and the chapter list page shows them all. No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed. Vector Algebra. SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works. - Ch 9 NCERT book - 80 marks, theory paper - 3 quick checks - Claiming a degree when derivatives appear inside non-polynomial functions . If derivatives occur inside functions like sin(dy / dx), e^(dy / dx), or log(dy / dx) and cannot be eliminated, the degree is not defined even though the order still exists. - Leaving out the arbitrary constant C during indefinite integration steps . Always add the arbitrary constant C immediately upon integrating both sides of a differential equation to ensure the general solution is complete. - Mixing up the variables in linear differential equations of type dx / dy . For dx / dy + P₁ x = Q₁, verify that P₁ and Q₁ are functions of y only, compute I.F. using ∫ P₁ dy, and multiply by x rather than y. - Forgetting to replace the temporary variable v with y / x . After integrating a homogeneous differential equation solved via y = v x, always substitute v = y / x back into the final expression. ## Key Links - [CBSE](https://www.swavid.com/cbse) - [Class 12](https://www.swavid.com/cbse/class-12) - [Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup) - [All 13 chapters](https://www.swavid.com/cbse/class-12/maths/chapters) - [NCERT Mathematics Part II (Reprint 2026-27)](https://ncert.nic.in/textbook.php) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/vector-algebra) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/continuity-and-differentiability) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/integrals) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/application-of-integrals) - [Class 12 Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup)