求微分方程的通解 (1-x^2)y"-xy'=2 要详细过程。。。。
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解：不显含y型，记y'=p,则y"=dp/dx=p'，
原微分方程可化为
(1-x^2)p'-xp=2
p'-x/(1-x^2)p=2/(1-x^2)
公式法得
p=[e^(∫x/(1-x^2)dx][C1+∫2/(1-x^2)[e^(∫-x/(1-x^2)dx]dx]
=e^(-1/2)ln(1-x^2)[C1+∫{2/(1-x^2)e^[(1/2)ln(1-x^2)]}dx]
=(1-x^2)^(-1/2)[C1+∫{[2/(1-x^2)](1-x^2)^(1/2)}dx]
=(1-x^2)^(-1/2)[C1+∫{[2/(1-x^2)]^(1/2)dx]
=(1-x^2)^(-1/2)[C1+2arcsinx]
即dy/dx=(1-x^2)^(-1/2)[C1+2arcsinx]
∫dy=∫(1-x^2)^(-1/2)[C1+2arcsinx]dx
y=(1/2)∫[C1+2arcsinx]d(C1+2arcsinx)
得y=(1/4)(C1+2arcsinx)^2+C2
原微分方程可化为
(1-x^2)p'-xp=2
p'-x/(1-x^2)p=2/(1-x^2)
公式法得
p=[e^(∫x/(1-x^2)dx][C1+∫2/(1-x^2)[e^(∫-x/(1-x^2)dx]dx]
=e^(-1/2)ln(1-x^2)[C1+∫{2/(1-x^2)e^[(1/2)ln(1-x^2)]}dx]
=(1-x^2)^(-1/2)[C1+∫{[2/(1-x^2)](1-x^2)^(1/2)}dx]
=(1-x^2)^(-1/2)[C1+∫{[2/(1-x^2)]^(1/2)dx]
=(1-x^2)^(-1/2)[C1+2arcsinx]
即dy/dx=(1-x^2)^(-1/2)[C1+2arcsinx]
∫dy=∫(1-x^2)^(-1/2)[C1+2arcsinx]dx
y=(1/2)∫[C1+2arcsinx]d(C1+2arcsinx)
得y=(1/4)(C1+2arcsinx)^2+C2
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