Unit-I: Basic Concepts and First Order Differential Equations

• 1. Basic Concepts of Differential Equations
  • 1.1 Origin of Differential Equations
  • 1.2 Order and Degree of a Differential Equation
  • 1.3 Linear and Nonlinear Differential Equations
• 2. Formation of Differential Equations
• 3. Differential Equations of First Order and First Degree
  • 3.1 Separable Equations
  • 3.2 Homogeneous Equations
  • 3.3 Equations Reducible to Homogeneous Form
  • 3.4 Exact Differential Equations
• 4. Summary of Solving Methods

1. Basic Concepts of Differential Equations

A differential equation is a mathematical equation that relates some function with its derivatives. In physical applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two.

Definition: An equation containing an independent variable, a dependent variable, and one or more derivatives of the dependent variable with respect to the independent variable is called a Differential Equation (D.E.).

1.1 Origin of Differential Equations

Differential equations arise naturally from mathematical modeling of physical, biological, chemical, and economic processes. Whenever a relationship involves a rate of change, it leads to a differential equation.

Real-World Examples:

• Newton's Law of Cooling: The rate of change of temperature T of an object is proportional to the difference between its own temperature and the surrounding temperature T_s:

  dT/dt = -k(T - T_s)

• Population Growth Model: The rate of growth of a population P is proportional to the population present at any time t:

  dP/dt = kP

• Simple Harmonic Motion (Physics): The motion of an oscillating mass on a spring is described by:

  d²x/dt² + ω²x = 0

1.2 Order and Degree of a Differential Equation

Understanding the order and degree is the first step in classifying and finding methods to solve any differential equation.

Order: The order of a differential equation is the order of the highest-order derivative appearing in the equation.

Degree: The degree of a differential equation is the power of the highest-order derivative, when the equation is written as a rational integral algebraic expression in its derivatives (i.e., after clearing any fractional powers or radicals of the derivatives, and ensuring derivatives do not appear inside transcendental functions like sine, cosine, logarithm, or exponentials).

Important Rule for Finding Degree: The differential equation must be expressible as a polynomial in terms of its derivatives. If the equation cannot be written as a polynomial in its derivatives, the degree is not defined.

Illustrative Examples:

1. Equation: dy/dx + 5y = e^x
   Highest derivative: dy/dx (first derivative).
   Power of highest derivative: 1.
   Order = 1, Degree = 1
2. Equation: d²y/dx² + 3(dy/dx)⁴ - 5y = 0
   Highest derivative: d²y/dx² (second derivative).
   Power of highest derivative: 1.
   Order = 2, Degree = 1 (Note: The power 4 is on the first derivative, which does not dictate the degree).
3. Equation: [1 + (dy/dx)²]^3/2 = k(d²y/dx²)
   Step to clear fractional exponent: Square both sides to eliminate the fractional exponent:

   [1 + (dy/dx)²]³ = k²(d²y/dx²)²

   
   Now, the highest derivative is d²y/dx² and its power is 2.
   Order = 2, Degree = 2
4. Equation: d²y/dx² + sin(dy/dx) = 0
   Here, the derivative dy/dx is inside a trigonometric function. This cannot be expanded as a polynomial of derivatives.
   Order = 2, Degree = Not Defined

Common Mistake: Confusing the power of a lower-order derivative with the degree of the entire equation. Always locate the highest derivative first, and look only at its power once the equation is rationalized.

1.3 Linear and Nonlinear Differential Equations

Differential equations are broadly classified based on whether the dependent variable and its derivatives behave linearly.

Linear Differential Equation: A differential equation is called linear if:

1. The dependent variable y and all its derivatives are of degree 1.
2. No product terms of the dependent variable and its derivatives (e.g., y(dy/dx) or (dy/dx)(d²y/dx²)) are present.
3. No transcendental functions (such as sin(y), e^y, ln(y)) of the dependent variable exist.

Nonlinear Differential Equation: Any differential equation that does not satisfy all the conditions of linearity is classified as a nonlinear differential equation.

2. Formation of Differential Equations

The formation of a differential equation involves finding the differential equation whose general solution is given by a relation between variables containing arbitrary constants.

Methodology:

1. Identify the number of independent arbitrary constants in the given relation. Let the number of constants be n.
2. Differentiate the equation successively n times with respect to the independent variable to obtain n additional equations.
3. Eliminate the arbitrary constants from the original equation and the n newly formed equations.
4. The resulting equation containing independent variables, dependent variables, and derivatives is the required differential equation. Its order will be exactly equal to n.

Fundamental Rule: The order of the differential equation is equal to the number of independent arbitrary constants present in its family of curves.

Step-by-Step Example 1:

Form the differential equation of the family of curves represented by:

y = A cos(x) + B sin(x)

Step 1: Identify the number of arbitrary constants. Here, there are two constants (A and B). Thus, the resulting differential equation must be of order 2.

Step 2: Differentiate once with respect to x:

dy/dx = -A sin(x) + B cos(x) --- (Equation 1)

Step 3: Differentiate a second time with respect to x:

d²y/dx² = -A cos(x) - B sin(x) --- (Equation 2)

Step 4: Factor out the negative sign in Equation 2:

d²y/dx² = -(A cos(x) + B sin(x))

d²y/dx² = -y

d²y/dx² + y = 0

Step-by-Step Example 2:

Form the differential equation of the family of circles passing through the origin and having centers on the x-axis.

Step 1: Set up the equation of the curves. The center of any such circle lies on the x-axis at (a, 0), and since it passes through the origin (0,0), its radius must be a. The equation of the circle is:

(x - a)² + y² = a²

x² - 2ax + a² + y² = a²

x² + y² = 2ax --- (Equation 1)

Step 2: Differentiate once with respect to x:

2x + 2y(dy/dx) = 2a

a = x + y(dy/dx)

Step 3: Eliminate a by substituting this expression back into Equation 1:

x² + y² = 2x(x + y(dy/dx))

x² + y² = 2x² + 2xy(dy/dx)

y² - x² = 2xy(dy/dx)

2xy(dy/dx) + x² - y² = 0

3. Differential Equations of First Order and First Degree

A first-order, first-degree differential equation can be written in the general form:

dy/dx = f(x, y)

M(x, y)dx + N(x, y)dy = 0

3.1 Separable Equations (Variables Separable)

A differential equation is separable if it can be written in such a way that all terms involving the dependent variable y can be grouped with dy, and all terms involving the independent variable x can be grouped with dx.

Standard Form: dy/dx = g(x)h(y)

Solution Method:

Step 1: Separate the variables by multiplying or dividing:

(1 / h(y)) dy = g(x) dx

Step 2: Integrate both sides directly:

∫ (1 / h(y)) dy = ∫ g(x) dx + C

Step-by-Step Example:

Solve: dy/dx = (1 + y²) / (1 + x²)

Step 1: Separate the variables by moving all y terms to the left and x terms to the right:

(1 / (1 + y²)) dy = (1 / (1 + x²)) dx

Step 2: Integrate both sides:

∫ (1 / (1 + y²)) dy = ∫ (1 / (1 + x²)) dx

tan^-1(y) = tan^-1(x) + C

tan^-1(y) - tan^-1(x) = C

tan^-1((y - x) / (1 + xy)) = C

(y - x) / (1 + xy) = tan(C) = C₁

y - x = C₁(1 + xy)

3.2 Homogeneous Equations

To understand homogeneous differential equations, we must first define a homogeneous function.

Homogeneous Function: A function f(x, y) is said to be homogeneous of degree n if, for any real number t:

f(tx, ty) = tⁿ f(x, y)

Homogeneous Differential Equation: A differential equation of the form dy/dx = f(x, y) is homogeneous if f(x, y) is a homogeneous function of degree 0. That is:

dy/dx = g(y/x)

Solution Method:

Step 1: Substitute y = vx. Here, v is a function of x.

Step 2: Differentiate y = vx with respect to x using the product rule:

dy/dx = v + x(dv/dx)

Step 3: Substitute y and dy/dx into the differential equation to obtain a new differential equation in variables v and x:

v + x(dv/dx) = g(v)

Step 4: Separate the variables x and v:

x(dv/dx) = g(v) - v

(1 / (g(v) - v)) dv = (1 / x) dx

Step 5: Integrate both sides to get the solution in terms of v and x:

∫ (1 / (g(v) - v)) dv = ln(x) + C

Step 6: Substitute v = y/x back into the solution to get the final answer in terms of x and y.

Step-by-Step Example:

Solve: (x² + y²)dx - 2xydy = 0

First, rewrite the equation in dy/dx form:

dy/dx = (x² + y²) / 2xy

Step 1 & 2: Substitute y = vx and dy/dx = v + x(dv/dx):

v + x(dv/dx) = (x² + (vx)²) / (2x(vx))

v + x(dv/dx) = x²(1 + v²) / (2vx²)

v + x(dv/dx) = (1 + v²) / 2v

Step 3: Isolate x(dv/dx):

x(dv/dx) = ((1 + v²) / 2v) - v

x(dv/dx) = (1 + v² - 2v²) / 2v

x(dv/dx) = (1 - v²) / 2v

Step 4: Separate variables:

(2v / (1 - v²)) dv = (1 / x) dx

Step 5: Integrate both sides:

∫ (2v / (1 - v²)) dv = ∫ (1 / x) dx

-ln|1 - v²| = ln|x| + ln|C|

ln|x| + ln|1 - v²| = ln|C₁| (where C₁ = 1/C)

x(1 - v²) = C₁

Step 6: Substitute v = y/x back:

x(1 - y²/x²) = C₁

x((x² - y²)/x²) = C₁

x² - y² = C₁x

3.3 Equations Reducible to Homogeneous Form

Equations of the form:

dy/dx = (a₁x + b₁y + c₁) / (a₂x + b₂y + c₂)

Case I: When a₁/a₂ ≠ b₁/b₂

In this case, we shift the origin of the coordinate system to eliminate the constants.

Step 1: Substitute x = X + h and y = Y + k, where h and k are constants to be determined. Since h and k are constants, dx = dX and dy = dY, so:

dY/dX = (a₁(X + h) + b₁(Y + k) + c₁) / (a₂(X + h) + b₂(Y + k) + c₂)

dY/dX = (a₁X + b₁Y + (a₁h + b₁k + c₁)) / (a₂X + b₂Y + (a₂h + b₂k + c₂))

Step 2: To make this equation homogeneous, choose h and k such that:

a₁h + b₁k + c₁ = 0

a₂h + b₂k + c₂ = 0

Step 3: The equation simplifies to the homogeneous form:

dY/dX = (a₁X + b₁Y) / (a₂X + b₂Y)

Step 4: After finding the solution in terms of X and Y, substitute X = x - h and Y = y - k to express the final answer in terms of original variables x and y.

Case II: When a₁/a₂ = b₁/b₂ = m

When the ratios of the coefficients are equal, the previous method fails because the equations for h and k will either have no solution or infinitely many solutions. We use an alternative substitution.

Let a₁/a₂ = b₁/b₂ = m. Then, a₁ = ma₂ and b₁ = mb₂. The differential equation becomes:

dy/dx = (m(a₂x + b₂y) + c₁) / ((a₂x + b₂y) + c₂)

Step 1: Substitute u = a₂x + b₂y.

Step 2: Differentiate with respect to x:

du/dx = a₂ + b₂(dy/dx)

dy/dx = (1 / b₂) (du/dx - a₂)

Step 3: Substitute this back into the differential equation:

(1 / b₂) (du/dx - a₂) = (mu + c₁) / (u + c₂)

3.4 Exact Differential Equations

A first-order differential equation written in the form M(x, y)dx + N(x, y)dy = 0 is called an exact differential equation if its left-hand side matches the exact differential of some function u(x, y).

Criterion for Exactness: The differential equation M(x, y)dx + N(x, y)dy = 0 is exact if and only if:

∂M/∂y = ∂N/∂x

where ∂ represents partial derivatives.

Step-by-Step Solution Method:

Once the equation is verified to be exact, its general solution is given by:

∫ M dx [treating y as constant] + ∫ (terms in N not containing x) dy = C

Step-by-Step Example:

Solve: (x³ + 3xy²)dx + (3x²y + y³)dy = 0

Step 1: Identify M and N.

M = x³ + 3xy²

N = 3x²y + y³

Step 2: Check the exactness condition. Compute the partial derivative of M with respect to y (treating x as constant):

∂M/∂y = ∂/∂y (x³ + 3xy²) = 0 + 6xy = 6xy

∂N/∂x = ∂/∂x (3x²y + y³) = 6xy + 0 = 6xy

Step 3: Apply the integration formula.
1. Integrate M with respect to x (treating y as a constant):

∫ M dx = ∫ (x³ + 3xy²) dx = x⁴/4 + (3/2)x²y²

∫ (terms in N free from x) dy = ∫ y³ dy = y⁴/4

Step 4: Combine the results. The general solution is:

x⁴/4 + (3/2)x²y² + y⁴/4 = C'

x⁴ + 6x²y² + y⁴ = C (where C = 4C')

Common Mistake: In step 3, students often integrate the whole of N instead of only selecting terms free of x. Remember, any term in N containing an x variable must be completely omitted from the second integral.

4. Summary of Solving Methods

This table serves as an exam quick-reference to determine which solving technique to apply based on the structure of the differential equation.