Q43P - Mathematical Methods in Physical Sciences

Question

Solve for all possible values of the real numbers  and in the following equations ${\left(x-iy\right)}^{\mathbf{2}}\mathbf{=}\mathbf{2}\mathit{i}\mathit{x}$.

Step-by-Step Solution

Answer

The answer is $\left(x,y\right)=\left(0,0\right) or \left(1,1\right) or \left(-1,1\right).$

Step 1: Given information

It is given that ${\left(x-iy\right)}^2=2ix.$ .

Step 2: Definition of a complex number

Every complex number can be described as: 

 

z=a+bi

Where a and b are both real numbers, z is the complex number, and i is known as iota, which makes z a complex number.

Step 3: Begin by stating the given information

State the given information as:

 

$$\begin{gathered} {\left(x-iy\right)}^2=2ix \\ {x}^2+2xyi-y^2=2xi \end{gathered}$$

Step 4: Equate like terms

Equate like terms from both sides:

 

$$\begin{gathered} x^2-y^2=0 \\ {x}^2=y^2 \\ and \\ x\left(y-1\right)=0 \end{gathered}$$

Step 5: Evaluate possible solutions

The above equation implies:

 $$\begin{gathered} If x=0 then y=0 \\ If y=1 then x=\pm 1 \end{gathered}$$

 

Thus, the final answer is $\left(x,y\right)=\left(0,0\right) or \left(1,1\right) or \left(-1,1\right)$ .