The given differential equation is:
dy/dx = y * e7x
To solve this first-order linear differential equation, we can use the technique of separation of variables. The main idea is to rearrange the equation in a way that allows us to separate variables y and x.
1. **Rearranging the equation**: We can separate the variables:
dy/y = e7x dx
2. **Integrating both sides**: Now, we integrate both sides to isolate y:
∫ (1/y) dy = ∫ e7x dx
The left side integrates to:
ln|y| + C1
And the right side integrates to:
(1/7)e7x + C2
3. **Combining results**: Equating both integrals gives us:
ln|y| = (1/7)e7x + C
where $C = C2 – C1$ is a constant.
4. **Exponentiating to solve for y**: We can exponentiate both sides to solve for y:
|y| = e(1/7)e7x + C = e(1/7)e7x} * eC
This simplifies to:
|y| = A * e(1/7)e7x} where $A = e^C$ is a positive constant.
Thus, we can write:
y = ± A * e(1/7)e7x}
5. **Final form of the general solution**: Recognizing that ±A can be represented as a new constant K, we arrive at the final general solution:
y = K * e(1/7)e7x}
where K is any real constant. This is the general solution of the differential equation.