Suppose that $y\left(x\right)$ solves the ordinary differential equation $\frac{dy}{dx}=3y$ with the initial condition $y\left(0\right)=2$. What is $y\left(x\right)$?

• A. $y\left(x\right)=3e^x$
• B. $y\left(x\right)=2e^{3x}$
• C. $y\left(x\right)=3e^{2x}$
• D. $y\left(x\right)=2e^x$

Answer: (B)

Verified step by step guidance

1. Step 1: Recognize that the given differential equation is a first-order linear ordinary differential equation of the form y' = ky, where k = 3 in this case. This type of equation has a standard solution form y(x) = Ce^(kx), where C is a constant determined by the initial condition.
2. Step 2: Substitute the given initial condition y(0) = 2 into the general solution y(x) = Ce^(kx). This will allow us to solve for the constant C.
3. Step 3: Evaluate y(0) = Ce^(3*0). Since e^(0) = 1, this simplifies to y(0) = C. Using the initial condition y(0) = 2, we find that C = 2.
4. Step 4: Substitute the value of C back into the general solution y(x) = Ce^(kx). This gives y(x) = 2e^(3x).
5. Step 5: Verify the solution by differentiating y(x) = 2e^(3x) to check that it satisfies the original differential equation y' = 3y. Differentiating, we find y'(x) = 6e^(3x), and substituting y(x) = 2e^(3x) into y' = 3y confirms that the equation holds true.