Worldwide Differential Equations with Linear Algebra by Robert McOwen
By Robert McOwen
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Extra info for Worldwide Differential Equations with Linear Algebra
In fact, we are interested in the case that there is a differentiable function Φ(x, y) so that M= ∂Φ ∂x and N = ∂Φ . 28) is exact and we call Φ a potential function for the vector field (M, N ). e. 28). Let us consider an example. Example 4. Suppose we want the general solution of 2x sin y dx + x2 cos y dy = 0. By inspection, we see that Φ(x, y) = x2 sin y has the desired partial derivatives ∂Φ = 2x sin y ∂x and ∂Φ = x2 cos y. ∂y Consequently, x2 sin y = c provides the general solution (in implicit form).
25) becomes dv + (1 − α) p(x) v = (1 − α) q(x). 26) to recover y. Let us perform a simple example. Example 3. Find the general solution of x2 y + 2 x y = 3 y 4 . Solution. 25) with α = 4: y + 2 3 y = 2 y4 . 27), it is better to derive the equation that v satisfies: v = y −3 ⇒ y = v −1/3 1 y = − v −4/3 v , 3 ⇒ 1 2 3 − v −4/3 v + v −1/3 = 2 v −4/3 , 3 x x and after some elementary algebra we obtain the linear equation ⇒ v − 6 9 v = − 2. x x As integrating factor, we take I(x) = e−6 x−1 dx = e−6 ln |x| = x−6 , which enables us to find v: (x−6 v) = −9 x−8 ⇒ x−6 v = 9 −7 x +C 7 ⇒ v= 9 −1 x + C x6 .
Theorem 2. 14) that are linearly independent on I. e. y(x) = c1 y1 (x) + c2 y2 (x) for all x ∈ I. 2. 14). To prove this theorem, we need to use linear algebra for a system of two equations in two unknowns x1 , x2 . 15) always admits the trivial solution x1 = 0 = x2 , but we want to know when it admits a nontrivial solution: at least one of x1 or x2 is nonzero. 16) has a solution x1 , x2 . Lemma 1. 15) admit a nontrivial solution (x1 , x2 ) if and only if ad − bc = 0. 16) admit a unique solution (x1 , x2 ) for each choice of (y1 , y2 ) if and only if ad − bc = 0.