1. Math
  2. Advanced Math
  3. consider the following equation x2 y sinxy ...

Question: consider the following equation x2 y sinxy ...

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3. Consider the following equation z y sin(ry) 0 (a) Prove that this equation has a unique continuously differentiable solution y y(r) such that y(0) 0 in a small neighborhood 0), and find y (0) (b) Prove that y y(z) is twice continuously differentiable in a small neighborhood of r 0, and find y(0). (c) Discuss the monotonicity of the function y y(a) near 0. (d) Does this equation have a unique solution z z(y) such that z (0) 0 near (0,0) Why? 4. Let f be a real valued function defined in a neighborhood UCR of (0,1), and assume that f E C1(U,R), f (0, 1) f 0 and f(0,1)-0. Prove that the equation sins ds has a unique, continuously differentiable solution t p(z) for (z, t) near (0, and express p (0) in terms of fr (0, 1) and fu(0, 1).

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