求常数a,b,使方程6Uxx-5Uxy+Uyy=0,在变量代换m=x+ay,n=x+by下,可化为新方程Umn=0偏导

Question

求常数a,b,使方程6Uxx-5Uxy+Uyy=0,在变量代换m=x+ay,n=x+by下,可化为新方程Umn=0
偏导

Answer 1

U'(x)=U'(m)m'(x)+U'(n)n'(x)=U'(m)+U'(n)；
U'(y)=U'(m)m'(y)+U'(n)n'(y)=aU'(m)+bU'(n)；
所以：U''(xx)=U''(mm)+2U''(mn)+U''(nn)；U''(xy)=aU''(mm)+bU''(mn)+aU''(mn)+bU''(nn)
U''(yy)=aU''(mm)a+aU''(mn)b+bU''(mn)a+bU''(nn)b
0=6Uxx-5Uxy+Uyy=6(U''(mm)+2U''(mn)+U''(nn))-5(aU''(mm)+bU''(mn)+aU''(mn)+bU''(nn))+
aU''(mm)a+aU''(mn)b+bU''(mn)a+bU''(nn)b
=(6-5a+a^2)U''(mm)+(12-5b-5a+2ab)U''(mn)+(6-5b+b^2)U''(nn)
由题意：6-5a+a^2=0,6-5b+b^2=0,解得：a=2,b=3或a=3,b=2或a=b=2或a=b=3