For the equation, , if y(0) = 3/7,then the value of y(1) is

To find the value of $y\left(1\right)$, we need to solve the given first-order separable differential equation shown in the image:

$\frac{dy}{dx}+7x^{2}y=0$

Step 1: Separate the variables
We can rewrite the differential equation by moving the term involving $y$ to the right-hand side:

$\frac{dy}{dx}=-7x^{2}y$

Assuming $y\ne 0$, divide both sides by $y$ and multiply by $dx$:

$\frac{1}{y}dy=-7x^{2}dx$

Step 2: Integrate both sides
Now, integrate both sides of the equation with respect to their respective variables:

$\int \frac{1}{y}dy=\int -7x^{2}dx$

Integrating gives:

$ln\left|y\right|=-\frac{7}{3}{x}^3+C$

where $C$ is the constant of integration. Taking the exponential of both sides, we get:

$y\left(x\right)=A{e}^{-\frac{7}{3}{x}^3}$

where $A=\pm e^C$ is a constant.

Step 3: Apply the initial condition
We are given the initial condition $y\left(0\right)=\frac{3}{7}$. Substitute $x=0$ and $y=\frac{3}{7}$ into the general solution:

$\frac{3}{7}=A{e}^{-\frac{7}{3}{\left(0\right)}^3}$

Since $e^0=1$, we find:

$A=\frac{3}{7}$

Thus, the particular solution is:

$y\left(x\right)=\frac{3}{7}{e}^{-\frac{7}{3}{x}^3}$

Step 4: Calculate y(1)
Substitute $x=1$ into the particular solution:

$y\left(1\right)=\frac{3}{7}{e}^{-\frac{7}{3}{\left(1\right)}^3}=\frac{3}{7}{e}^{-7/3}$

This matches Option 2 shown in the second image.