Maharashtra State Board · Class 12 · Mathematics · Part II Chapter 6

Differential Equations — Formula Sheet

Board Formulas

15 formulas

1.Order and Degree
$(\frac{d ^{2} y}{d x ^{2}})^{3} + (\frac{d y}{d x})^{4} + y = 0 \Rightarrow order 2, degree 3$
Order = order of the highest derivative present. Degree = power of that highest-order derivative, once the equation is a polynomial in the derivatives.
2.Degree After Clearing Radicals
$[1 + (\frac{d y}{d x})^{2}]^{3/2} = k \frac{d ^{2} y}{d x ^{2}} \Rightarrow [1 + (\frac{d y}{d x})^{2}]^{3} = k^{2} (\frac{d ^{2} y}{d x ^{2}})^{2}$
Remove fractional powers first (here, square both sides): order 2, degree 2. If a derivative sits inside sin, log, e^( ) etc., the degree is not defined.
3.Formation of a Differential Equation
$n arbitrary constants \Rightarrow differentiate n times \Rightarrow DE of order n$
Differentiate the given relation as many times as there are arbitrary constants, then eliminate the constants between the equations.
4.General and Particular Solution
$y = A cos x + B sin x is the general solution of \frac{d ^{2} y}{d x ^{2}} + y = 0$
A general solution has as many arbitrary constants as the order. A particular solution is obtained by fixing the constants from given conditions.
5.Variables Separable★
$\frac{d y}{d x} = \frac{f ( x )}{g ( y )} \Rightarrow \int g (y) d y = \int f (x) d x + c$
Collect all y-terms with dy and all x-terms with dx, then integrate both sides. Only one constant is needed.
6.Reducible to Variables Separable
$\frac{d y}{d x} = f (a x + b y + c) : put a x + b y + c = v, \frac{d y}{d x} = \frac{1}{b} (\frac{d v}{d x} - a)$
Example: dy/dx = (x + y)². Put x + y = v, so dv/dx = 1 + v², which separates.
7.Homogeneous Function
$f (λ x, λ y) = λ^{n} f (x, y)$
f is homogeneous of degree n. dy/dx = f(x, y)/g(x, y) is a homogeneous equation when f and g are homogeneous of the same degree.
8.Homogeneous Equation: Substitution★
$y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}$
After substituting, x cancels from the right-hand side and the equation separates in v and x. Replace v by y/x at the end.
9.Linear Differential Equation
$\frac{d y}{d x} + P y = Q$
$P$ : Coefficient of y, a function of x · $Q$ : Right-hand side, a function of x
P and Q are functions of x only (or constants). y and dy/dx appear only to the first power and are not multiplied together.
10.Integrating Factor
$I.F. = e^{\int P d x}$
Find P only after the coefficient of dy/dx is 1. Simplify with e^(log f(x)) = f(x).
11.Solution of a Linear Equation★
$y (I.F.) = \int Q (I.F.) d x + c$
The integrating factor appears on BOTH sides. Integrate the right-hand side carefully — it often needs integration by parts.
12.Linear Equation in x
$\frac{d x}{d y} + P x = Q \Rightarrow x (I.F.) = \int Q (I.F.) d y + c, I.F. = e^{\int P d y}$
Use it when the equation is linear in x rather than y; here P and Q are functions of y.
13.Law of Natural Growth and Decay★
$\frac{d x}{d t} = k x \Rightarrow x = x_{0} e^{k t}$
$x$ : Amount present at time t · $x_{0}$ : Initial amount (at t = 0) · $k$ : Constant of proportionality (per unit time)
k > 0 for growth (population, bacteria), k < 0 for decay (radioactive substance). x₀ is the amount at t = 0.
14.Half-Life
$x_{0} e^{- k T} = \frac{x _{0}}{2} \Rightarrow T = \frac{l o g 2}{k}$
For decay written as dx/dt = −kx with k > 0. The half-life does not depend on the initial amount. log is the natural logarithm.
15.Newton's Law of Cooling★
$\frac{d θ}{d t} = - k (θ - θ_{0}) \Rightarrow θ - θ_{0} = C e^{- k t}$
$θ$ : Temperature of the body at time t (°C) · $θ_{0}$ : Temperature of the surroundings (°C), constant · $k$ : Positive constant (per unit time)
The rate of cooling is proportional to the excess of the body's temperature over its surroundings. C is the initial excess temperature, θ(0) − θ₀.