We are given the first-order differential equation:

$x(x + y)y' + y(3x + y) = 0$

Our goal is to solve this equation.

Step 1: Rearrange the equation

First, divide the entire equation by $x$, assuming $x \neq 0$:

$(x + y)y' + \frac{y(3x + y)}{x} = 0$

Step 2: Simplify and inspect

Rewrite the terms more clearly:

$(x + y)y' + y \left(\frac{3x}{x} + \frac{y}{x}\right) = 0$ $(x + y)y' + y(3 + \frac{y}{x}) = 0$

Step 3: Change of variables

To reduce the complexity, we can try the substitution $v = \frac{y}{x}$, which implies $y = vx$. Differentiate both sides with respect to $x$:

$y' = v + x \frac{dv}{dx}$

Substitute $y = vx$ and $y' = v + x \frac{dv}{dx}$ into the equation:

$(x + vx)\left(v + x \frac{dv}{dx}\right) + vx(3 + v) = 0$

Step 4: Simplify the equation

Expand the terms:

$x(v + 1)\left(v + x \frac{dv}{dx}\right) + vx(3 + v) = 0$ $x(v + 1)v + x(v + 1)x \frac{dv}{dx} + vx(3 + v) = 0$ $x(v + 1)x \frac{dv}{dx} + x \left( (v + 1)v + v(3 + v) \right) = 0$ Now simplify the second part: $(v + 1)v + v(3 + v) = v^2 + v + 3v + v^2 = 2v^2 + 4v$ Thus, the equation becomes:

$x(v + 1)x \frac{dv}{dx} + x(2v^2 + 4v) = 0$ $x^2(v + 1) \frac{dv}{dx} + x(2v^2 + 4v) = 0$

Step 5: Separate variables

We can divide through by $x$, assuming $x \neq 0$:

$x(v + 1) \frac{dv}{dx} + (2v^2 + 4v) = 0$

Rearrange it:

$x(v + 1) \frac{dv}{dx} = -(2v^2 + 4v)$

Now divide both sides by $v + 1$:

$x \frac{dv}{dx} = -\frac{2v^2 + 4v}{v + 1}$

Step 6: Simplify the right-hand side

Simplify the expression on the right:

$\frac{2v^2 + 4v}{v + 1} = 2v$

So the equation reduces to:

$x \frac{dv}{dx} = -2v$

Step 7: Solve the differential equation

Separate the variables $v$ and $x$:

$\frac{dv}{v} = -\frac{2}{x} dx$

Integrate both sides:

$\ln |v| = -2 \ln |x| + C_1$

Simplify the right-hand side:

$\ln |v| = \ln \left(\frac{C_2}{x^2}\right)$

Exponentiate both sides:

$|v| = \frac{C_2}{x^2}$

Thus:

$v = \frac{C_2}{x^2}$

Since $v = \frac{y}{x}$, we have:

$\frac{y}{x} = \frac{C_2}{x^2}$

Step 8: Solve for $y$

Multiply both sides by $x$:

$y = \frac{C_2}{x}$

Thus, the general solution to the differential equation is:

$y = \frac{C}{x}$ where $C = C_2$ is a constant.

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Would you like more details on any part of the solution?

Related Questions:

1. How does substitution help in solving differential equations?
2. What are the general steps to solve separable differential equations?
3. How does the assumption $x \neq 0$ affect the solution?
4. Could a different substitution method be used here?
5. What are other common methods to solve first-order differential equations?

Tip:

In solving differential equations, substitution is a powerful tool for simplifying complex expressions and reducing the equation to a form that can be more easily integrated.