工程數學~一階方程式求解

1. y' + 3y = 8

積分因子 = exp(S3dx) = e^(3x)

e^3x (y' + 3y) = 8e^3x

(ye^3x)' = 8e^3x

ye^3x = 8/3 e^3x + C

y = 8/3 + Ce^-3x，C是一常數


2. y' = x/y

用分離變數法(separation of variables)

S y dy = S x dx

y^2/2 = x^2/2 + C

y^2 - x^2 = C', C和C'是常數

y(1) = 3

(3)^2 - (1)^2 = C'

C' = 8

所以，y^2 - x^2 = 8


3. 9yy' + 4x = 0

9yy' = -4x

用分離變數法

9 S y dy = -4 S x dx

9y^2/2 = -2x^2 + C

4x^2 + 9y^2 = C', C和C'是常數


4. y' = -2xy

用分離變數法

S 1/y dy = -2 S x dx

lny = -x^2 + C

y = Ae^(-x^2), A和C是常數

y(0) = 1

A = 1

所以, y = e^(-x^2)

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