To find the general solution for the differential equation dy/dx = y * e^(5x), we will use the method of separation of variables. This method allows us to rearrange the equation so that each variable (y and x) is on its own side.
First, we can rearrange the equation:
dy/dx = y * e^(5x)
We start by separating the variables:
1/y dy = e^(5x) dx
Next, we integrate both sides:
Integrating the left side with respect to y gives:
∫ 1/y dy = ln|y| + C1
Integrating the right side with respect to x gives:
∫ e^(5x) dx = (1/5)e^(5x) + C2
Combining these results, we get:
ln|y| = (1/5)e^(5x) + C
Where C is a constant that combines both constants of integration C1 and C2.
To solve for y, we exponentiate both sides:
|y| = e^( (1/5)e^(5x) + C)
We can rewrite this as:
|y| = e^C * e^( (1/5)e^(5x))
Letting K = e^C, we can express this as:
y = K * e^( (1/5)e^(5x))
Where K is an arbitrary constant. Thus, the general solution to the differential equation dy/dx = y * e^(5x) is:
y = C * e^( (1/5)e^(5x))
where C is any real constant. This represents the family of solutions to the differential equation.