Knowledge organisers

Differential Equations: CBSE Class 12 Maths knowledge organiser

Everything to know about differential equations on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Order: The order of the highest derivative in a differential equation.
Degree: The power of the highest-order derivative, when the equation is a polynomial in the derivatives.
General solution: Has as many arbitrary constants as the order; a particular solution fixes them.

Key formulas

Order = highest derivative; degree = power of it (when polynomial in derivatives). General solution has as many constants as the order
Variables separable	\(\int\frac{dy}{g(y)}=\int f(x)\,dx+C\)
Homogeneous \(\frac{dy}{dx}=F\big(\frac yx\big)\)	\(y=vx,\) \(\frac{dy}{dx}=v+x\frac{dv}{dx}\)
Linear \(\frac{dy}{dx}+Py=Q\)	\(\text{I.F.}=e^{\int P\,dx},\) \(y\cdot\text{I.F.}=\int Q\cdot\text{I.F.}\,dx+C\)
Linear \(\frac{dx}{dy}+P_1x=Q_1\)	\(\text{I.F.}=e^{\int P_1\,dy},\) \(x\cdot\text{I.F.}=\int Q_1\cdot\text{I.F.}\,dy+C\)

Worked example

\(\dfrac{dy}{dx} = xy\): \(\ln|y| = \dfrac{x^2}{2} + c \Rightarrow y = Ce^{x^2/2}\).

Common mistakes

Stating the degree without first clearing a radical or fractional power.
Forgetting to write the equation in standard form (coefficient of \(\dfrac{dy}{dx}\) equal to \(1\)) before finding \(P\).
Writing \(e^{\int P\,dx}\) with a “\(+C\)” inside — the IF needs no constant.
Adding the constant only after dividing: in \(y \cdot \text{IF} = \int Q\cdot\text{IF}\,dx + C\), \(C\) must be divided by the IF too.

You should be able to…

Find the order and degree of a differential equation and solve simple variables-separable equations.
Recognise and solve homogeneous and linear equations and find particular solutions from given conditions.
Write complete long answers and case studies, including setting up a differential equation from a real situation.
Handle equations that are linear in x, mixed-method problems and modelling questions that need interpretation.

The printable sheet

Differential Equations knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Differential Equations knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Matrices · Determinants · Continuity and Differentiability · Application of Derivatives · Integrals · Application of Integrals · Vector Algebra · Three Dimensional Geometry · Linear Programming · Probability · All CBSE Class 12 Maths organisers