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3) Prove by an application of the existence and uniqueness theorem that the functions x = c1 ϕ1 (t) + c2 ϕ2 (t), t ∈ R, c1 , c2 ∈ R, are the complete solution of (4). A. Linear diﬀerential equation of second order and of constant coeﬃcients, where the characteristic polynomial has a double root. D. Write the characteristic polynomial in two ways and compare. Insert ϕ2 (t) and apply the existence and uniqueness theorem. I. 1) It follows from R2 + a1 R + a0 = (R − r)2 = R2 − 2r R + r 2 , that a1 = −2r and a0 = r2 , hence 1 r = − a1 .

By the angle ϕ. Then choose the polar coordinate system in such a way that ϕ = 0 or ϕ = π. Then write c 1 c1 instead of ± c21 + c22 , where we choose the minus sign, if ϕ = π, and choose K > 0 and e = . K Then we get the following r= 1 1 1 = · , K + c1 cos θ K 1 + e cos θ where the excentricity e must not be confused with the number e. 2) Assume that |e| < 1. Then 1 + e cos θ > 0 for all θ. com 44 Calculus Analyse 1c-5 Linear differential equations og second order and of constant coefficients and hence by a squaring, (11) x2 + y 2 = 1 2e x + e2 x2 .

The Power of Knowledge Engineering Plug into The Power of Knowledge Engineering. 35 (1) Find the complete solution of the diﬀerential equation d2 u + u = K, dθ2 where (r, θ) are the polar coordinates, and u = 1/r. Prove that r as a function of θ is given by (10) r = 1 . K + c1 cos θ + c2 sin θ We shall in the following only consider the case c2 = 0. It can be proved that this can also be achieved by choosing the coordinate system properly. We shall now identify the curves which are described by the equation (10).