https://r2.erweima.ai/mathbot/image/1727192298989f0dmakwa.jpg

The image contains three differential equations with initial conditions. To find their particular solutions, I will solve each equation step by step.

---

### **(1) \((3xy + 2)dx + x^2 dy = 0, \ y(1) = 1\)**

This is a **first-order linear differential equation** in the form:

\[
(3xy + 2)dx + x^2 dy = 0
\]

To solve, rewrite the equation as:

\[
\frac{dy}{dx} = -\frac{3xy + 2}{x^2}
\]

Separate variables:

\[
\frac{dy}{y} = -\frac{3x + \frac{2}{y}}{x^2y+steps}
The general solution to the first differential equation \((3xy + 2)dx + x^2dy = 0\) is:

\[
y(x) = \frac{C_1}{x^2} - \frac{1}{x}
\]

Now, apply the initial condition \(y(1) = 1\) to find the particular solution.

\[
1 = \frac{C_1}{1^2} - \frac{1}{1}
\]
\[
1 = C_1 - 1
\]
\[
C_1 = 2
\]

Thus, the particular solution is:

\[
y(x) = \frac{2}{x^2} - \frac{1}{x}
\]

---

### **(2) \(\frac{dy}{dx} + y \cot(x) = 5e^{\cos(x)}, \ y\left(\frac{\pi}{2}\right) = -4\)**

This is a linear first-order differential equation. We can solve it using an integrating factor method.

We start with the standard form:

\[
\frac{dy}{dx} + P(x)y = Q(x)
\]

where \(P(x) = \cot(x)\) and \(Q(x) = 5e^{\cos(x)}\). Let's proceed with solving it.

Find the particular solutions to three differential equations using methods such as separation of variables and integrating factors. Initial conditions are applied to obtain unique solutions. Suitable for college-level calculus or differential equations courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation