We say that a solution to a differential equation is non-trivial if it is NOT identically...

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We say that a solution to a differential equation is non-trivial if it is NOT identically zero everywhere. Let K> 0 be a constant. (1a) Find two linearly independent solutions to the following differential equation and hence write down its general solution: y"(x) = xy(x) for x € (0, 1). (1b) Suppose the above differential equation is supplemented with one of the following boundary conditions: * Dirichlet: y(0) = 0 and y(1) = 0 * Neumann: y'(0) = 0 and y'(1) = 0 * Mixed-I: y'(0) = 0 and y(1) = 0 * Mixed-II: y(0) = 0 and y'(1) = 0 For which of the above boundary conditions, the general solution found in Question (la) remains non- trivial? Justify your answer. We say that a solution to a differential equation is non-trivial if it is NOT identically zero everywhere. Let K> 0 be a constant. (1a) Find two linearly independent solutions to the following differential equation and hence write down its general solution: y"(x) = xy(x) for x € (0, 1). (1b) Suppose the above differential equation is supplemented with one of the following boundary conditions: * Dirichlet: y(0) = 0 and y(1) = 0 * Neumann: y'(0) = 0 and y'(1) = 0 * Mixed-I: y'(0) = 0 and y(1) = 0 * Mixed-II: y(0) = 0 and y'(1) = 0 For which of the above boundary conditions, the general solution found in Question (la) remains non- trivial? Justify your answer.

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