Non-Homogeneous Differential Equation (Linearly Independent)

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

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 independent case: $a_1 b_2 - a_2 b_1 \neq 0$ (i.e. $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$)
  • Solved by translating coordinates to eliminate constant terms: substitute $x = u + h$, $y = v + k$.

Examples Covered

  1. Example 1: $(y - x - 2),dx + (4y + x - 3),dy = 0$
  2. Example 2: $(x + y),dx + (x - y + 2),dy = 0$ — solution is $y - 1 - 2(x+1)(y-1) - (x+1)^2 = A$
  3. Example 3: $(2x - 5y + 3),dx - (2x + 4y - 6),dy = 0$ with $y(1) = 2$
  4. Example 4: $(7y - 3),dx + (2x + 1),dy = 0$ with $y(1) = 2$

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