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標題:
發問:
Show that Pinney's equation d^2y/dx^2 + y = c/y^3 is satisfied by y = [ u^2 +cv^2/W^2 ]^0.5 where u and v are independent solutions of d^2z/dx^2 + z = 0 and W= uv' - vu' = constant , c being an arbitrary constant.
最佳解答:
CKW is extremely bad The question should be d^2y/dx^2 - y = c/y^3 u and v are independent solutions of d^2z/dx^2 - z = 0 Now let u=e^x and v=e^(-x) W=uv' - vu' = -2 = constant y=(e^(2x)+(c/4)e^(-2x))^(1/2) y^2=e^(2x)+(c/4)e^(-2x) 2yy'=2e^(2x)-(c/2)e^(-2x) y'=(1/y)[e^(2x)-(c/4)e^(-2x)] y'' =(1/y^2){y[2e^(2x)+(c/2)e^(-2x)]-y'[e^(2x)-(c/4)e^(-2x)]} =(1/y)[2e^(2x)+(c/2)e^(-2x)]-(1/y^3)[e^(2x)-(c/4)e^(-2x)] =(2/y)[y^2]-(1/y^3){[e^(2x)+(c/4)e^(-2x)]^2-c} =2y-(1/y^3)(y^4-c) =y+c/y^3
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differential equation發問:
Show that Pinney's equation d^2y/dx^2 + y = c/y^3 is satisfied by y = [ u^2 +cv^2/W^2 ]^0.5 where u and v are independent solutions of d^2z/dx^2 + z = 0 and W= uv' - vu' = constant , c being an arbitrary constant.
最佳解答:
CKW is extremely bad The question should be d^2y/dx^2 - y = c/y^3 u and v are independent solutions of d^2z/dx^2 - z = 0 Now let u=e^x and v=e^(-x) W=uv' - vu' = -2 = constant y=(e^(2x)+(c/4)e^(-2x))^(1/2) y^2=e^(2x)+(c/4)e^(-2x) 2yy'=2e^(2x)-(c/2)e^(-2x) y'=(1/y)[e^(2x)-(c/4)e^(-2x)] y'' =(1/y^2){y[2e^(2x)+(c/2)e^(-2x)]-y'[e^(2x)-(c/4)e^(-2x)]} =(1/y)[2e^(2x)+(c/2)e^(-2x)]-(1/y^3)[e^(2x)-(c/4)e^(-2x)] =(2/y)[y^2]-(1/y^3){[e^(2x)+(c/4)e^(-2x)]^2-c} =2y-(1/y^3)(y^4-c) =y+c/y^3
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