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Given an IVP Fundamental Existence Theorem for Linear Differential Equations d" y an (x)- dx" + an-1(x) + α₁ (x) ª + dy
Posted: Thu Jun 30, 2022 7:34 pm
by answerhappygod
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Given an IVP Fundamental Existence Theorem for Linear Differential Equations d" y an (x)- dx" + an-1(x) + α₁ (x) ª + dy dx ... d-ly dxn-1 y(xo) = yo, y (xo) = y₁, , -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 € I that exists on the whole interval I. Consider the IVP on the whole real line (x²-121) dªy d³ y +x². + dx4 dx³ y(12) = 4, y (12) = 12, + ao(x) y = g(x) 1 dy x² + 121 dx "(12) = 1, y" (12) = 8, + y = sin(x)
The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval