Answer:
To solve the first-order ordinary differential equation \[ y' = 3y^2(x + \cos(x)) \] with the initial condition \(y(0) = -2\), we can use the method of separation of variables. Here are the steps to solve this differential equation:
1. Rewrite the equation in the form \(\frac{dy}{dx} = 3y^2(x + \cos(x))\).
2. Separate the variables by moving \(y^2\) terms to the left side and \(x + \cos(x)\) terms to the right side:
\[\frac{dy}{y^2} = 3(x + \cos(x))dx\]
3. Integrate both sides:
\[\int \frac{1}{y^2} dy = 3\int (x + \cos(x)) dx\]
Let's solve each integral separately:
For the left integral:
\[\int \frac{1}{y^2} dy = -\frac{1}{y} + C_1\]
For the right integral, you'll need to integrate \(x\) and \(\cos(x)\) separately:
\[\int (x + \cos(x)) dx = \int x dx + \int \cos(x) dx\]
Integrate \(x\) with respect to \(x\):
\[\int x dx = \frac{1}{2}x^2 + C_2\]
Integrate \(\cos(x)\) with respect to \(x\):
\[\int \cos(x) dx = \sin(x) + C_3\]
Now, combine these results:
\[-\frac{1}{y} + C_1 = \frac{1}{2}x^2 + \sin(x) + C_2 + C_3\]
4. Combine the constants \(C_1\), \(C_2\), and \(C_3\) into a single constant \(C\):
\[-\frac{1}{y} + C = \frac{1}{2}x^2 + \sin(x) + C_4\]
5. Solve for \(y\):
\[-\frac{1}{y} = \frac{1}{2}x^2 + \sin(x) + C_4 - C\]
Now, multiply both sides by -1:
\[\frac{1}{y} = -\frac{1}{2}x^2 - \sin(x) - (C_4 - C)\]
6. Take the reciprocal of both sides:
\[y = \frac{1}{-\frac{1}{2}x^2 - \sin(x) - (C_4 - C)}\]
7. Simplify the denominator:
\[y = \frac{1}{-\frac{1}{2}x^2 - \sin(x) + (C - C_4)}\]
Now, let's use the initial condition \(y(0) = -2\) to find the value of the constant \(C\):
\[y(0) = \frac{1}{-\frac{1}{2}(0)^2 - \sin(0) + (C - C_4)} = \frac{1}{0 + 0 + (C - C_4)}\]
Since \(y(0) = -2\), we have:
\[-2 = \frac{1}{C - C_4}\]
Now, solve for \(C - C_4\):
\[-2 = \frac{1}{C - C_4}\]
Multiply both sides by \(C - C_4\):
\[-2(C - C_4) = 1\]
Now, isolate \(C - C_4\):
\[C - C_4 = -\frac{1}{2}\]
So, \(C = C_4 - \frac{1}{2}\).
Now, substitute this expression for \(C\) back into the solution for \(y\):
\[y = \frac{1}{-\frac{1}{2}x^2 - \sin(x) + (C_4 - \frac{1}{2})}\]
Finally, the solution for the initial value problem is:
\[y = \frac{1}{-\frac{1}{2}x^2 - \sin(x) + (C_4 - \frac{1}{2})}\]
where \(C_4\) is any constant.
Explanation: