dy/dx = 7y, and y = 1 when x = 0. solve the equation. choose 1 answer: a y = 7e^7x b y = e^7x c y = e^x d y…

Explanation:

Step1: Separate variables

We have $\frac{dy}{dx}=7y$. Rearranging gives $\frac{dy}{y}=7dx$.

Step2: Integrate both sides

Integrating $\int\frac{dy}{y}=\int7dx$. The left - hand side integral is $\ln|y|$ and the right - hand side is $7x + C$, so $\ln|y|=7x + C$.

Step3: Solve for y

Exponentiating both sides, we get $y = e^{7x + C}=e^{C}e^{7x}$. Let $A = e^{C}$, then $y = Ae^{7x}$.

Step4: Use the initial condition

Given $y = 1$ when $x = 0$. Substituting into $y = Ae^{7x}$, we have $1=Ae^{0}$, so $A = 1$. Then $y=e^{7x}$.

Answer:

B. $y = e^{7x}$