This chapter introduces differential equations, including their types and applications across various scientific fields.
Differential Equations – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - II, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Differential Equations chapter of Mathematics Part - II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
dy/dx = g(x)
This expresses that the derivative of y with respect to x is equal to a function g of x. It represents the basic form of a differential equation.
Order of a differential equation: n
The order n is defined as the highest derivative in the equation. For example, in y'' + y' + y = 0, the order is 2.
Degree of a differential equation: m
The degree m is the highest power of the highest order derivative in polynomial form. E.g., in (dy/dx)² + y = 0, the degree is 2.
General solution: y = f(x) + C
The general solution includes arbitrary constant C. It's a family of curves representing the solution for all initial conditions.
Particular solution: y = f(x, C₀)
A particular solution is obtained by specifying values for arbitrary constants in the general solution.
Separation of variables: ∫(1/h(y)) dy = ∫g(x) dx
This method involves separating terms involving y and x, integrating both sides to solve the differential equation.
Integrating Factor (I.F): e^(∫P(x)dx)
An integrating factor is used to convert a non-exact differential equation into an exact one, thereby facilitating its solution.
Homogeneous differential equation: dy/dx = F(x,y)
An equation is homogeneous if F(λx, λy) = λ^n F(x, y) for degree n. The general solution often involves substitution.
Exact equation: M(x,y)dx + N(x,y)dy = 0
This is an equation that can be expressed as the total differential of a function. If M_y = N_x, it is exact.
Linear first-order differential equation: dy/dx + P(x)y = Q(x)
This form represents a linear relationship in which y and its derivatives are of the first degree.
Equations
d²y/dx² + p dy/dx + qy = 0
A second-order linear differential equation representing systems in equilibrium in physics, such as oscillating springs.
dy/dx = k y
Represents exponential growth or decay, where k is a constant. Common in population models and finance.
dx/dt = ax + by
A system of ordinary differential equations, often used in modeling coupled systems in physics.
dy/dx = (y - x)/(x + y)
A differential equation that could represent the relationship between two variables in a reaction rate scenario.
y' + p(x)y = q(x)
A linear first-order differential equation, where p(x) and q(x) are functions of x, used commonly in applications.
M(x,y) + N(x,y) = 0
A condition for exact equations, where both M and N are functions of x and y.
∫(dy/y) = ∫k dx
Logarithmic solution of a first-order separable differential equation illustrating growth or decay processes.
y = C e^(ax)
This represents the general solution of a linear constant-coefficient differential equation. C is the constant.
F(x,y) = 0
Represents the implicit formulation of solutions, useful for geometrical interpretations of curves.
dy/dx = f(x,y)
A general representation where the slope at any point depends on the current position and can be solved via various methods.
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