A differential equation is an equation that contains a function and its derivatives. It is widely used in physics, chemistry, biology, pharmacy and engineering to describe change.
Order and Degree of a Differential Equation
Order
The order of a differential equation is the highest order derivative present.
Degree
The degree is the power of the highest order derivative after the equation is made free from fractions and radicals.
Examples
- dy/dx + y = 0 → order = 1, degree = 1
- d²y/dx² + 3(dy/dx) = 0 → order = 2, degree = 1
Formation of Differential Equations
To form a differential equation, differentiate the given equation repeatedly and eliminate constants.
Solution of Differential Equations
Solutions may contain arbitrary constants. They are of two types:
1. General Solution
A solution containing arbitrary constants.
2. Particular Solution
Obtained by substituting initial or boundary conditions into the general solution.
First-Order and First-Degree Differential Equations
1. Variable Separable Method
Equation of the form:
dy/dx = f(x) g(y)
Separate variables:
dy/g(y) = f(x) dx
Integrate both sides.
2. Homogeneous Differential Equations
An equation dy/dx = f(x, y) is homogeneous if f(tx, ty) = f(x, y).
Substitute y = vx → dy/dx = v + x dv/dx. Solve the resulting equation.
3. Linear Differential Equations
Standard form:
dy/dx + Py = Q
Integrating Factor (IF)
IF = e∫P dx
Solution
y(IF) = ∫(Q · IF) dx + C
4. Exact Differential Equations
Equation of the form:
M(x, y) dx + N(x, y) dy = 0
Condition for Exactness
∂M/∂y = ∂N/∂x
Solution Method
Integrate M with respect to x. Add terms whose derivative is included in N. Equate to constant.
5. Linear Equation in x (Reverse Form)
Sometimes equation is written as:
dx/dy + P₁x = Q₁
Solve similarly using integrating factor.
Differential Equations Reducible to Linear Form
Equations like:
dy/dx + P(x)y = Q(x) yⁿ
Use Bernoulli’s equation substitution:
v = y¹⁻ⁿ
Clairaut’s Differential Equation
Form: y = x dy/dx + f(dy/dx)
Solution contains both general solution and singular solution.
Applications of Differential Equations
- Population growth and decay
- Radioactive decay
- Chemical kinetics processes
- Drug elimination and absorption models
- Motion under gravity
- Newton’s law of cooling
Examples (Concept-Based)
- Solving dy/dx = (x² + 1)/(y² + 1) using variable separation
- Solving homogeneous equations using substitution y = vx
- Solving linear differential equations using IF
- Checking exactness by verifying ∂M/∂y = ∂N/∂x
- Applying Bernoulli equation substitution
Practice Questions
- Solve dy/dx = (3x²)/(2y)
- Solve dy/dx + 4y = e⁻ˣ
- Check if (3x²y + y²) dx + (x³ + 2xy) dy = 0 is exact
- Solve dy/dx + y tan x = sin x
- Form differential equation of family y = A e²ˣ + B e⁻ˣ
Detailed Notes:
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