神马作文网多元微积分问题
来源：学生作业帮 编辑：神马作文网作业帮 分类：数学作业 时间：2024/11/18 23:01:24
多元微积分问题
(i) Zxx=d^2z/dx^2,Zyy=d^2y/dy^2
z(x,y)=f(u)=f(e^x*siny)
dz/dx=f'(u)*du/dx=f'(u)*e^x*siny
Zxx=d^2z/dx^2=f''(u)e^xsiny*e^x*siny+f'(u)*e^x*siny
dz/dy=f'(u)*du/dy=f'(u)*e^x*cosy
Zyy=d^2z/dy^2=f''(u)e^xcosy*cosy*e^x+f'(u)(-siny)*e^x
(ii) Zxx+Zyy=e^(2x)*Z
{f''(u)e^xsiny*e^x*siny+f'(u)*e^x*siny}
+{f''(u)e^xcosy*cosy*e^x+f'(u)(-siny)*e^x}
=e^(2x)*Z=e^(2x)*f(u)
=> f''(u)=f(u) => f''(u)-f(u)=0
此为二阶齐次微分方程,特征方程为 r^2-1=0
特征根为 r1,2=±1
∴其通解为 f(u)=C1e^u+C2e^(-u)
f'(u)=C1e^u-C2e^(-u)
带入初始条件f(0)=0,f'(0)=1,可得
f(0)=C1+C2=0,f'(0)=C1-C2=1
可解得 C1=1/2,C2=-1/2
∴f(u)=1/2*e^u-1/2*e^(-u)
z(x,y)=f(u)=f(e^x*siny)
dz/dx=f'(u)*du/dx=f'(u)*e^x*siny
Zxx=d^2z/dx^2=f''(u)e^xsiny*e^x*siny+f'(u)*e^x*siny
dz/dy=f'(u)*du/dy=f'(u)*e^x*cosy
Zyy=d^2z/dy^2=f''(u)e^xcosy*cosy*e^x+f'(u)(-siny)*e^x
(ii) Zxx+Zyy=e^(2x)*Z
{f''(u)e^xsiny*e^x*siny+f'(u)*e^x*siny}
+{f''(u)e^xcosy*cosy*e^x+f'(u)(-siny)*e^x}
=e^(2x)*Z=e^(2x)*f(u)
=> f''(u)=f(u) => f''(u)-f(u)=0
此为二阶齐次微分方程,特征方程为 r^2-1=0
特征根为 r1,2=±1
∴其通解为 f(u)=C1e^u+C2e^(-u)
f'(u)=C1e^u-C2e^(-u)
带入初始条件f(0)=0,f'(0)=1,可得
f(0)=C1+C2=0,f'(0)=C1-C2=1
可解得 C1=1/2,C2=-1/2
∴f(u)=1/2*e^u-1/2*e^(-u)