Unit 5: Higher Order DEs & PDEs

1. Introduction: General Solution

This unit focuses on linear DEs of order 'n' with constant coefficients:

General Solution of a Non-Homogeneous DE

The complete solution (General Solution) is the sum of two parts:

y = y_c + y_p

• y_c (Complementary Function):
  The general solution of the corresponding homogeneous equation (i.e., setting R(x) = 0). It will contain 'n' arbitrary constants.
• y_p (Particular Integral):
  A *specific* solution (with no arbitrary constants) of the full non-homogeneous equation.

2. Homogeneous Linear DEs with Constant Coefficients (Finding y_c)

To find the Complementary Function (y_c), we solve the homogeneous equation:

Where D is the differential operator (D = d/dx, D² = d²/dx², etc.)

Step-by-Step Procedure:

1. Form the Auxiliary Equation (AE):
   Replace 'D' with a variable 'm' and set the polynomial to zero.
   a_nmⁿ + ... + a₁m + a₀ = 0
2. Find the roots (m₁, m₂, ..., m_n) of the AE.
3. Write y_c based on the roots:

3. Non-Homogeneous Linear DEs: The Particular Integral (PI)

To find the Particular Integral (y_p), we "solve" for y from the equation f(D)y = R(x).

Here, 1/f(D) is an "inverse operator". The method depends on the form of R(x).

4. Different Forms of Particular Integrals

This is a set of "short-cut" rules for finding y_p.

Form 1: R(x) = e^ax

Rule: Replace every 'D' with 'a'.

Case of Failure: If f(a) = 0, this means (D-a) is a factor of f(D).
Rule: If f(D) = (D-a)^kg(D), where g(a)≠0, then:

Form 2: R(x) = sin(ax) or cos(ax)

Rule: Replace every 'D²' with '(-a²)'. (Do *not* replace D with -a).

Case of Failure: If f(-a²) = 0.
Rule: The method is similar to the "Case of Failure" for e^ax, but more complex. A common trick is to use y_p = x · [solution] (for a single failure).

Form 3: R(x) = x^m (a polynomial)

Rule: 1. Rewrite [ 1 / f(D) ] using algebraic manipulation (e.g., long division or binomial theorem) to get a series in *ascending* powers of D.
(e.g., 1 / (1-D) = 1 + D + D² + ... ) 2. Apply this operator series to x^m. Since D³(x²) = 0, the series will terminate.

Form 4: R(x) = e^ax · V(x) (where V is any function, e.g., sin(x) or x²)

Rule (Shift Theorem): 1. "Shift" the e^ax to the front. 2. Replace every 'D' in the operator with '(D + a)'. 3. Apply the new operator [ 1 / f(D+a) ] to V(x) using one of the other rules.

Form 5: R(x) = x · V(x) (where V is sin(x), cos(x), etc.)

Rule: This is a general rule. 1. First find the PI for V alone: W = [ 1 / f(D) ] · V. 2. Then the full PI is: y_p = x·W - [ f'(D) / (f(D))² ] · V

5. Formation of Partial Differential Equations (PDEs)

A PDE is an equation involving partial derivatives. We form them by eliminating arbitrary elements (constants or functions).

Standard Notation:
z = f(x, y)
p = ∂z/∂x
q = ∂z/∂y

1. By Elimination of Arbitrary Constants

If an equation has arbitrary constants, you must differentiate partially w.r.t. x and y to get enough equations to eliminate the constants.

Example: Form a PDE from z = ax + by + ab

1. ∂z/∂x = a => p = a
2. ∂z/∂y = b => q = b
3. Substitute 'a' and 'b' back into the original equation.

2. By Elimination of Arbitrary Functions

If an equation has an arbitrary function (e.g., f(...)), you must eliminate the *function itself*.

Example: Form a PDE from z = f(x + y)

1. Let u = x + y. So z = f(u).
2. Find p and q using the chain rule:
3. p = ∂z/∂x = (df/du) · (∂u/∂x) = f'(u) · 1 = f'(u)
4. q = ∂z/∂y = (df/du) · (∂u/∂y) = f'(u) · 1 = f'(u)
5. We have p = f'(u) and q = f'(u). This means p and q are equal.

6. Solution of Partial Differential Equations (Lagrange's Method)

This section covers solving first-order, linear PDEs, which are in Lagrange's Form:

Lagrange's Method of Solution:

1. Form the "Subsidiary Equations" (or "Auxiliary Equations"):
   This is a set of symmetric ordinary differential equations.
   dx / P = dy / Q = dz / R
2. Find two independent solutions:
   Take any two ratios (e.g., dx/P = dy/Q) and solve them. This gives a first solution:
   u(x, y, z) = c₁
   
   Then, take another pair (or use multipliers) to find a *second*, *independent* solution:
   v(x, y, z) = c₂
3. Write the General Solution:
   The general solution can be written in two equivalent forms, where F (or Φ) is an arbitrary function.
   F(u, v) = 0
   or
   u = F(v)