Given An Ivp Fundamental Existence Theorem For Linear Differential Equations D Ly Dx 1 A X Dx Ao X G X D 1 (31.71 KiB) Viewed 51 times
Given An Ivp Fundamental Existence Theorem For Linear Differential Equations D Ly Dx 1 A X Dx Ao X G X D 2 (22.46 KiB) Viewed 51 times
Given an IVP Fundamental Existence Theorem for Linear Differential Equations d-ly dx"-1 .+ a₁(x) dx + ao(x) = g(x) ... dx y(xo) = yo, y (xo) = y₁, .., y(n-1)(x) = Yn-1 If the coefficients a,, (x), ..., ao (x) and the right hand side of the equation g(x) are continuous on an interval I and if a,, (x) #0 on I then the IVP has a unique solution for the point xo E I that exists on the whole interval I. Consider the IVP on the whole real line dªy d" y an (x)- dx" +an-1(x). + +x4 d³y (x²-49). 1 dy dx4 dx³ x² + 49 dx y(0) = -360, y (0) = 20, y'(0) = 4, y" (0) = 7, + + y = sin(x)
The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval
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