Differential Equation Solution Explained

This problem asks us to find the value of \(y(1)\) for a given first-order differential equation \(\frac{{dy}}{{dx}} + 7{x^2}y = 0\) with an initial condition \(y(0) = \frac{3}{7}\). This is a separable differential equation, which means we can separate the variables \(y\) and \(x\) on opposite sides of the equation.

Step-by-Step Differential Equation Solution

Let's solve the given differential equation step-by-step:

1. Rearrange the Differential Equation:

   The given differential equation is:

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

   First, move the term involving \(y\) to the right side of the equation:

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

   Now, separate the variables by moving all terms involving \(y\) to one side and all terms involving \(x\) to the other side:

   \[\frac{{dy}}{y} = -7{x^2}dx\]
2. Integrate Both Sides:

   Next, integrate both sides of the separated equation. Remember that \(\int \frac{1}{y} dy = \ln|y|\) and \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\).

   \[\int \frac{{dy}}{y} = \int -7{x^2}dx\] \[\ln|y| = -7 \left(\frac{x^{2+1}}{2+1}\right) + C\] \[\ln|y| = -7 \frac{x^3}{3} + C\] \[\ln|y| = -\frac{7}{3}x^3 + C\]

   To solve for \(y\), we exponentiate both sides (raise \(e\) to the power of both sides):

   \[|y| = e^{-\frac{7}{3}x^3 + C}\] \[|y| = e^{-\frac{7}{3}x^3} \cdot e^C\]

   Let \(A = \pm e^C\). Since the initial condition \(y(0) = \frac{3}{7}\) is positive, we can assume \(y\) will remain positive for the solution, so \(y = A e^{-\frac{7}{3}x^3}\).

   \[y(x) = A e^{-\frac{7}{3}x^3}\]
3. Apply the Initial Condition to Find the Constant \(A\):

   We are given the initial condition \(y(0) = \frac{3}{7}\). Substitute \(x=0\) and \(y=\frac{3}{7}\) into our general solution \(y(x) = A e^{-\frac{7}{3}x^3}\):

   \[\frac{3}{7} = A e^{-\frac{7}{3}(0)^3}\] \[\frac{3}{7} = A e^0\]

   Since \(e^0 = 1\), we get:

   \[\frac{3}{7} = A \cdot 1\] \[A = \frac{3}{7}\]
4. Form the Particular Solution:

   Now substitute the value of \(A\) back into the general solution to get the particular solution for this initial value problem:

   \[y(x) = \frac{3}{7}e^{-\frac{7}{3}x^3}\]
5. Calculate the Value of \(y(1)\):

   Finally, we need to find the value of \(y(1)\). Substitute \(x=1\) into the particular solution:

   \[y(1) = \frac{3}{7}e^{-\frac{7}{3}(1)^3}\] \[y(1) = \frac{3}{7}e^{-\frac{7}{3}}\]

Comparison with Options

Let's compare our calculated value of \(y(1)\) with the given options:

Our calculated value \(y(1) = \frac{3}{7}e^{-\frac{7}{3}}\) matches option 3.