110年 - 成大_資訊聯招_工程數學#105389

(a)

(b)

2. (10%) Find the value of m, so that the given differential equation is exact.
(6xy³+cosy)dx+(2mx²y²-xsiny)dy=0

3. (10%) In the following terms, which does not exist the Laplace transform?

4. (10%) Given the Laplace transform of Bessel function Please derive the Laplace transform of Bessel function J₀(3t).

5. (10%) Please find the indicial equation of the following differential equation.
y"+p(x)y'+q(x)y=0,
x = 0 is a regular sigular point.
where xp(x) = A₀ + A₁x+A2x²+...

x²g(x)= B₀ +B₁x+B₂x²+...

6. (10%) Find the serial solutions at x=0 for the following differential equation.

x(x-1)y"+(3x-l)y'+y=0

7. (10%) For Legendre's different equation (1-x²)y"-2xy'+λy =0 in which -1≤x≤1, and 2 is a real constant, please find the serial solutions at x=0 and explain how to get the Legendre's polynomials.

8. (10%) Find fit) from the following equation.

9. (10%) Solve the following equation
(10%)

110年 - 成大_資訊聯招_工程數學#105389

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