Find the General Solution of the Given Differential Equation dydx = y e^(7x)

To solve the differential equation dy/dx = y e^7x, we can use the method of separation of variables.

First, rewrite the equation to separate the variables:

dy/y = e^7x dx

Now, integrate both sides:

∫(1/y) dy = ∫e^7x dx

The left side integrates to:

ln|y|

The right side requires a bit more attention. The integral of e^7x is:

(1/7)e^7x + C, where C is the constant of integration.

Putting it all together, we have:

ln|y| = (1/7)e^7x + C

To eliminate the natural logarithm, we exponentiate both sides:

|y| = e^{(1/7)e7x + C}

This simplifies to:

y = ± e^{(1/7)e7x} * eC}

Finally, let K = ±e^C, which is just another constant. Thus, we express the general solution as:

y = K e^(1/7)e7x

So, the general solution of the given differential equation is:

y = K e^(1/7)e7x