设函数f(u)具有二阶导数,而z=f((e^x)*sin(y))满足方程d^2(z)/d^2(x^2)+d^2(z)/d(y^2)=e^(2*x)*z,求f(u).

题目

设函数f(u)具有二阶导数,而z=f((e^x)*sin(y))满足方程d^2(z)/d^2(x^2)+d^2(z)/d(y^2)=e^(2*x)*z,求f(u).

答案

令u=e^x*siny,则z=f(u)
∂z/∂x=∂z/∂u*∂u/∂x=f'(u)*e^x*siny=uf'(u),∂²z/∂x²=∂(uf'(u))/∂x=uf'(u)+u²f''(u)
∂z/∂y=f'(u)*e^x*cosy,∂²z/∂y²=∂(f'(u)*e^x*cosy)/∂y=f''(u)*e^(2x)*cos²y-f'(u)*e^x*siny=f''(u)*e^(2x)*cos²y-uf'(u)
故∂²z/∂x²+∂²z/∂y²=uf'(u)+u²f''(u)+f''(u)*e^(2x)*cos²y-uf'(u)=u²f''(u)+f''(u)*e^(2x)*cos²y=f''(u)*[e^(2x)*sin²y+e^(2x)*cos²y]=f''(u)*e^(2x)=e^(2x)*z
所以有f''(u)=z=f(u),积分可得：f(u)=C1e^u+C2e^(-u) (C1、C2为任意常数)