By Lucia M.
We think about the equationIf Ω is of sophistication , we convey that this challenge has a non-trivial resolution u λ for every λ ∊ (8π, λ*). the price λ* will depend on the area and is bounded from less than by way of 2 j zero 2 π, the place j zero is the 1st 0 of the Bessel functionality of the 1st type of order 0 (λ*≥ 2 j zero 2 π > eight π). additionally, the relations of answer u λ blows-up as λ → eight π.
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Extra resources for A blowing-up branch of solutions for a mean field equation
However, this is not a purely mathematical concern, for at least two reasons. If a problem has no solution, we would prefer to know that fact before investing time and effort in a vain attempt to solve the problem. Further, if a sensible physical problem is modeled mathematically as a differential equation, then the equation should have a solution. If it does not, then presumably there is something wrong with the formulation. In this sense an engineer or scientist has some check on the validity of the mathematical model.
8) contains two terms and that the first term is part of the result of differentiating the product µ(t)y. Thus, let us try to determine µ(t) so that the left side of Eq. (8) becomes the derivative of the expression µ(t)y. If we compare the left side of Eq. (8) with the differentiation formula dy dµ(t) d [µ(t)y] = µ(t) + y, dt dt dt (9) we note that the first terms are identical and that the second terms also agree, provided we choose µ(t) to satisfy dµ(t) = 2µ(t). dt (10) Therefore our search for an integrating factor will be successful if we can find a solution of Eq.
20), dy + ay = g(t), dt whose solutions are given by Eq. (23), y = e−at eas g(s) ds + ce−at . The solutions converge if a > 0, as in Example 2, and diverge if a < 0, as in Example 3. 2, however, Eq. (20) does not have an equilibrium solution. The final stage in extending the method of integrating factors is to the general first order linear equation (3), dy + p(t)y = g(t), dt where p and g are given functions. If we multiply Eq. (3) by an as yet undetermined function µ(t), we obtain dy + p(t)µ(t)y = µ(t)g(t).