Find the solution to the differential equation $\frac{dy}{dx}=3y$ with the initial condition $y\left(0\right)=2$.

• A. $y=2e^x$
• B. $y=3e^{2x}$
• C. $y=2e^{3x}$
• D. $y=6e^x$

Answer: (C)

Verified step by step guidance

1. Step 1: Recognize that the given differential equation dy/dx = 3y is separable, meaning it can be rewritten in the form (1/y) dy = 3 dx.
2. Step 2: Integrate both sides of the equation. For the left-hand side, integrate (1/y) dy, which results in ln|y|. For the right-hand side, integrate 3 dx, which results in 3x + C, where C is the constant of integration.
3. Step 3: Solve for y by exponentiating both sides to remove the natural logarithm. This gives y = e^(3x + C). Using properties of exponents, rewrite this as y = e^C * e^(3x), where e^C is a constant.
4. Step 4: Apply the initial condition y(0) = 2 to determine the value of the constant e^C. Substitute x = 0 and y = 2 into the equation y = e^C * e^(3x), which simplifies to 2 = e^C * e^0. Since e^0 = 1, this gives e^C = 2.
5. Step 5: Substitute the value of e^C back into the equation to get the final solution y = 2e^(3x).