Exact Differential Equation

Lecture slides covering exact differential equations and their solution method.

Key Points

  • A first-order DE $M(x,y),dx + N(x,y),dy = 0$ is exact when: $$ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} $$
  • Solution found by integrating $M$ w.r.t. $x$ and $N$ w.r.t. $y$, then combining to find $F(x,y) = C$.

Examples Covered

  1. Example 1: $(3x^2 y + 2x^2 - x),dx + (x^3 + 2y^3 - 2y),dy = 0$ — show exact and solve.
  2. Example 2: $(3x + y - 1),dy + (2x^3 + 3y),dx = 0$ — show solution is $6xy + y^2 - 2y + x^4 = A$.
  3. Example 3: $(xy\cos xy + \sin xy),dx + (x^2\cos xy + e^y),dy = 0$ — exact with $y(0) = 1$.

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