As a matter of fact, I used the Lagrangian funtion.
L=X1^a*X2^b-λ(P1 X1+P2 X2-M)
The λ is called Lagrangian Multiplier. Take the derivatives w.r.t X1 and X2 and λ respectively, you will have three equations, these are called First-Order Conditions (FOCs).
  FOCs:
L(X_1 )=aX1^(a-1)*X2^b-λP1=0   ❶
L(X_2 )=bX1^a*X_2^(b-1)-λP2=0       ❷
Lλ=-(P1 X1+P2 X2-M)=0    ❸
Solve the system of these three equations, you will get the solutions.
X1(P1, P2, M) =aM/((a+b) P1 )
X2(P1, P2, M) =  bM/((a+b) P2 )
The λ is the marginal utility of income.
This is the standard procedure of utility maximization problem.

Oh~~I get it.   It's kind of you to write the detailed processes for me. Thank you very much.

U R welcome!

不好意思，好久没上了，才看到你的回复，是的，这是科布道格拉斯函数的推导过程，要构建拉格朗日函数：L=clnx1+clnx2-λ（p1x1+p2x2-m）然后微分
dL/dX1=c/x1-λp1=0 dL/dx2=d/x2-λp2=0 dL/dλ=p1x1+p2x2-m=0解得c=λp1x1 d=λp2x2 所以c+d=λ(p1x1+p2x2)=λm可得λ=m/c+d将此式带入c与d解得x1与x2的需求函数结论就是x1=cm/(c+d)p1
x2=dm/(c+d)p2,c和d分别代表购买商品x1与x2所占收入的比例