Unit-1: Ordinary Differential Equations of Higher Order

Unit-1: Ordinary Differential Equations of Higher Order

1. Linear Differential Equation of nth Order with Constant Coefficients

General form:

\[ a_n \frac{d^ny}{dx^n} + a_{n-1} \frac{d^{n-1}y}{dx^n} + \cdots + a_1 \frac{dy}{dx} + a_0 y = X(x) \]

Auxiliary equation for homogeneous part (\(X(x) = 0\)):

\[ a_n m^n + a_{n-1} m^{n-1} + \cdots + a_1 m + a_0 = 0 \]

Complementary Function (CF) depends on roots:

  • Real and distinct roots \(m_1, m_2, \dots, m_n\):
    • \[ y_c = c_1 e^{m_1 x} + c_2 e^{m_2 x} + \cdots + c_n e^{m_n x} \]
  • Repeated root \(m\) of multiplicity \(k\):
    • \[ y_c = (c_1 + c_2 x + c_3 x^2 + \cdots + c_k x^{k-1}) e^{m x} \]
  • Complex roots \(\alpha \pm i\beta\):
    • \[ y_c = e^{\alpha x} (c_1 \cos \beta x + c_2 \sin \beta x) \]

Particular Integral (PI) for different forms of \(X(x)\):

\(X(x)\)Form of PI
\(e^{ax}\)\(y_p = A e^{ax}\)
(unless \(a\) is a root → multiply by \(x^k\))
\(\sin ax\) or \(\cos ax\)\(y_p = A \cos ax + B \sin ax\)
Polynomial of degree \(n\): \(p(x)\)\(y_p = a_n x^n + \cdots + a_0\)
(multiply by \(x^k\) if 0 is root \(k\) times)
\(x^m e^{ax}\)\(y_p = x^k (a_m x^m + \cdots + a_0) e^{ax}\)
\(e^{ax} (A \cos bx + B \sin bx)\)\(y_p = x^k e^{ax} (C \cos bx + D \sin bx)\)

2. Simultaneous Linear Differential Equations

Example system:

\[ \frac{dx}{dt} + P x + Q y = f_1(t) \] \[ \frac{dy}{dt} + R x + S y = f_2(t) \]

Operator method (using \(D = \frac{d}{dt}\)):

\[ (D + P)x + Q y = f_1(t) \] \[ R x + (D + S)y = f_2(t) \]

Solve using elimination or substitution after treating as algebraic equations in \(D\).

3. Second-Order Linear DE with Variable Coefficients

General form:

\[ \frac{d^2 y}{dx^2} + P(x) \frac{dy}{dx} + Q(x) y = R(x) \]

4. Solution by Changing Independent Variable

Used when equation lacks \(y\) or \(x\):

  • If missing \(y\) → let \(v = \frac{dy}{dx}\), then \(\frac{d^2 y}{dx^2} = v \frac{dv}{dy}\)
  • If missing \(x\) → let \(v = \frac{dy}{dx}\), then \(\frac{d^2 y}{dx^2} = \frac{dv}{dx} = \frac{dv}{dy} \cdot \frac{dy}{dx} = v \frac{dv}{dy}\)

5. Method of Variation of Parameters (2nd Order)

For \( y'' + P y' + Q y = R(x) \), assume:

\[ y_p = u_1 y_1 + u_2 y_2 \] where \(y_1, y_2\) are fundamental solutions.

System:

\[ u_1' y_1 + u_2' y_2 = 0 \] \[ u_1' y_1' + u_2' y_2' = R(x) \]

Solution using Wronskian \(W = y_1 y_2' - y_2 y_1'\):

\[ u_1' = -\frac{y_2 R(x)}{W}, \quad u_2' = \frac{y_1 R(x)}{W} \] \[ u_1 = \int -\frac{y_2 R}{W} dx, \quad u_2 = \int \frac{y_1 R}{W} dx \]

6. Cauchy-Euler (Equidimensional) Equation

Standard form:

\[ a_n x^n \frac{d^n y}{dx^n} + a_{n-1} x^{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1 x \frac{dy}{dx} + a_0 y = X(x) \]

Substitution: let \( x = e^t \) → \( t = \ln x \), then:

\[ y = y(t), \quad x \frac{dy}{dx} = \frac{dy}{dt}, \quad x^2 \frac{d^2 y}{dx^2} = \frac{d^2 y}{dt^2} - \frac{dy}{dt} \]

For homogeneous 2nd order:

\[ a x^2 y'' + b x y' + c y = 0 \] Auxiliary equation: \[ a m(m-1) + b m + c = 0 \]

Cases:

  • Roots \(m_1 \neq m_2\): \( y = c_1 x^{m_1} + c_2 x^{m_2} \)
  • Repeated root \(m\): \( y = (c_1 + c_2 \ln x) x^m \)
  • Complex roots \(\alpha \pm i\beta\): \( y = x^\alpha (c_1 \cos(\beta \ln x) + c_2 \sin(\beta \ln x)) \)

7. Applications in Engineering

  • Mechanical Vibrations (Free): \( m \frac{d^2 x}{dt^2} + k x = 0 \) → \( \omega = \sqrt{k/m} \)
  • Damped Vibrations: \( m \ddot{x} + c \dot{x} + k x = 0 \)
    Discriminant: \( \Delta = c^2 - 4mk \)
  • Forced Vibrations: \( m \ddot{x} + c \dot{x} + k x = F_0 \cos \omega t \)
  • RLC Circuit: \( L \frac{d^2 q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C} q = E(t) \)
  • Beam Deflection: \( EI \frac{d^4 y}{dx^4} = w(x) \)
Back to Math-2