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4. Find the eigenvalues and eigenfunctions of the following problem. (20%)
y"+λy=0；y'(0)=y'(L)=0

其他申論題

2. Prove the integral as following (15%)

(a) Find the eigenvalues (λ1 ≥ λ2 ≥ λ3) of the matrix (10%)

(b) Obtain an orthogonal matrix P which givesPTAP=diag (λ1,λ2,λ3) (10%)

5. Determine the inverse Laplace transform of the function. (15%)

6. Let vectorfield F=2yzi-4xzjtspk. Evaluate the value of the surface integral , where Σis the sphere of radius 5 about (-1,3,1) . (15%)

For problem 1~10, each counts 10 points 1.

4. Please prove the quotient rule: If f and g are differentiable at x and g(x)≠0, then the quotient f /g is differentiable at x and

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