Differential Equations — Class 12 Maharashtra HSC Mathematics revision notes
A 4-page visual revision sheet for Differential Equations, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Differential Equations in one look
an equation built from derivatives, solved by separating, substituting, or multiplying by one integrating factor
Order and Degree
- a differential equation involves a function and its derivatives, solving it means finding the function
- order: the highest order of derivative that appears
- degree: the power of that highest derivative, written with no fractional or negative powers
- a derivative trapped inside a square root, log or sine has no defined degree until cleared
Order, Degree and Formation
- order: the index of the highest derivative present
order=n for y^(n) - degree: its power, once rationalised to polynomial form
degree = p in left(y^(n)right)^p - a family with n arbitrary constants needs n differentiations to remove them all
n constants ⇒ order n - differentiate n times, then eliminate the constants between the relation and its derivatives, leaving
Φ(x,y,y',…,y^(n))=0
Order and degree, the special cases
- a derivative sits under a radical, log or trig function — degree is undefined until the equation is cleared to polynomial form
- a family carries n independent arbitrary constants — the differential equation formed has order n
- every term shares the same total degree in x and y — the equation is homogeneous, substitute y=vx
- P and Q are functions of y instead of x — use the sideways linear form
Reading off order and degree
Q. Find the order and degree of x^2d^2y/dx^2+3xleft(dy/dxright)^2+4y=0.
- highest order derivative present: d^2y/dx^2, second order
- it appears to the power 1, so degree is 1
- the lower-order (dy/dx)^2 term plays no part in either count
Answer: order 2, degree 1
Forming a differential equation
Q. Form the differential equation of the family y=Ae^2x, A an arbitrary constant.
- differentiate once: dy/dx = 2Ae^2x
- the original gives Ae^2x=y, substitute directly
- dy/dx = 2y
Answer: dy/dx - 2y = 0
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Differential Equations sheet covers
- Order and Degree
- Order, Degree and Formation
- order: the index of the highest derivative present
- degree: its power, once rationalised to polynomial form
- a family with n arbitrary constants needs n differentiations to remove them all
- differentiate n times, then eliminate the constants between the relation and its derivatives, leaving
- Order and degree, the special cases
- Reading off order and degree
- Forming a differential equation
- Variable Separable Form
- Solving a variable separable equation
- Homogeneous Differential Equations
- Homogeneous equation via y=vx
- Variable separable vs linear
- Variable separable
- Linear, dy/dx+Py=Q
- Linear Differential Equation
- standard form, P and Q functions of x
- integrating factor
- general solution, multiplying through by the I.F.
- Solving a linear differential equation
- Which method actually applies
- Applications: Growth, Decay and Cooling
- Exponential growth from a proportional rate
- Newton's law of cooling
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Differential Equations the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.