The Solution of the Ordinary Differential Equation dy/dx = y, y(0) = 0

To solve the given ordinary differential equation (ODE) dy/dx = y with the initial condition y(0) = 0, we can use the method of separation of variables. This method involves separating the variables x and y on either side of the equation and integrating both sides.

Step 1: Separation of Variables
To begin, we separate the variables by moving all the y terms to one side and all the x terms to the other side:
dy/y = dx

Step 2: Integrating Both Sides
Next, we integrate both sides of the equation:
∫(dy/y) = ∫dx

The integral of 1/y with respect to y is ln|y| + C1, where C1 is the constant of integration. The integral of dx with respect to x is simply x + C2, where C2 is another constant of integration. Therefore, the equation becomes:
ln|y| + C1 = x + C2

Step 3: Combining Constants of Integration
Since C1 and C2 are arbitrary constants, we can combine them into a single constant C:
ln|y| = x + C

Step 4: Solving for y
To solve for y, we need to eliminate the natural logarithm. We can do this by taking the exponential of both sides:
e^(ln|y|) = e^(x + C)

The exponential of the natural logarithm cancels out, leaving us with:
|y| = e^x * e^C

Since e^C is just another constant, we can rewrite this as:
|y| = Ce^x

Step 5: Considering the Initial Condition
To satisfy the initial condition y(0) = 0, we substitute x = 0 into the equation:
|0| = Ce^0

This simplifies to:
0 = C

Therefore, the constant C is equal to 0, and the equation becomes:
|y| = 0

Step 6: Final Solution
Since |y| = 0, the absolute value of y must be 0. Thus, the solution to the given ODE with the initial condition y(0) = 0 is:
y = 0