Non-Homogeneous Differential Equation (Linearly Dependent)

Lecture slides covering non-homogeneous first-order DEs where the linear terms are linearly dependent.

Key Points

  • Non-homogeneous DE: $M(x,y),dx + N(x,y),dy = 0$ with: $$ M(x,y) = a_1 x + b_1 y + c_1, \quad N(x,y) = a_2 x + b_2 y + c_2 $$
  • Linearly dependent case: $a_1 b_2 - a_2 b_1 = 0$ (i.e. $\frac{a_1}{a_2} = \frac{b_1}{b_2}$)
  • Solved by substituting $u = a_1 x + b_1 y$ to reduce to a separable equation.

Examples Covered

  1. Example 1: $(x + y),dx + (3x + 3y - 4),dy = 0$
  2. Example 2: $(x + y - 2),dx + (x + y + 2),dy = 0$ — simplifies to $(x + y + 2)^2,dx = 8x + A$
  3. Example 3: $(x - 2y + 3),dx - (4y - 2x + 1),dy = 0$ with $x = 0, y = 1$
  4. Example 4: $(x + y),dx + (x + y + 1),dy = 0$ with $y(2) = 1$

Links