Let us solve the problems one by one:

1. Solve the differential equation $\frac{dy}{dx} = -2x \tan y$, given $y(0) = \frac{\pi}{2}$:

Solution:

The equation is separable. Rewrite it as: $\frac{1}{\tan y} \, dy = -2x \, dx$

Now, simplify $\frac{1}{\tan y}$ to $\cot y$: $\cot y \, dy = -2x \, dx$

Integrate both sides: $\int \cot y \, dy = \int -2x \, dx$

The integral of $\cot y$ is $\ln|\sin y|$, and the integral of $-2x$ is $-x^2 + C$ (where $C$ is the constant of integration). Thus: $\ln|\sin y| = -x^2 + C$

Exponentiate both sides to solve for $\sin y$: $\sin y = e^{-x^2 + C} = e^C e^{-x^2}$

Let $e^C = k$ (a constant), so: $\sin y = k e^{-x^2}$

Using the initial condition $y(0) = \frac{\pi}{2}$: $\sin\left(\frac{\pi}{2}\right) = 1 \quad \text{implies} \quad k e^0 = k = 1$

Thus, the solution is: $\sin y = e^{-x^2}$

2. Determine the general solution of the first-order differential equation $\frac{dy}{dx} - \frac{2y}{x} = -7$ using integrating factors:

Solution:

This is a first-order linear differential equation. The standard form is: $\frac{dy}{dx} + P(x)y = Q(x)$ Here, $P(x) = -\frac{2}{x}$ and $Q(x) = -7$.

The integrating factor is given by: $\mu(x) = e^{\int P(x) \, dx} = e^{\int -\frac{2}{x} \, dx} = e^{-2 \ln|x|} = e^{\ln|x|^{-2}} = |x|^{-2}$

Multiply through by $\mu(x) = \frac{1}{x^2}$: $\frac{1}{x^2} \frac{dy}{dx} - \frac{2}{x^3} y = -\frac{7}{x^2}$

Simplify: $\frac{d}{dx} \left( \frac{y}{x^2} \right) = -\frac{7}{x^2}$

Integrate both sides: $\frac{y}{x^2} = \int -\frac{7}{x^2} \, dx = 7 \cdot \frac{1}{x} + C$

Simplify: $\frac{y}{x^2} = -\frac{7}{x} + C$

Multiply through by $x^2$: $y = -7x + Cx^2$

Thus, the general solution is: $y = -7x + Cx^2$

Let me know if you have questions or want more detailed explanations.

5 Follow-up Questions:

1. What are the applications of separable differential equations in physics?
2. Can the solution for $\sin y = e^{-x^2}$ be written in terms of $y$ explicitly?
3. How does the integrating factor simplify solving linear differential equations?
4. What are the limitations of using the integrating factor method?
5. How can initial conditions modify the general solution of a differential equation?

Tip:

For first-order linear differential equations, always check if rewriting in the standard form simplifies identifying the integrating factor directly.