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1. Solve: dy/dx = y/x + x * sin(y/x)

Solution:

We use the substitution ( v = y/x ), so ( y = v * x ).

Using the product rule:

dy/dx = v + x * dv/dx

Substitute into the original equation:

v + x * dv/dx = v + x * sin(v)

Simplify:

x * dv/dx = x * sin(v)

Divide both sides by ( x ):

dv/dx = sin(v)

This is a separable differential equation. Separate the variables:

dv / sin(v) = dx

Integrate both sides:

∫ (1 / sin(v)) dv = ∫ 1 dx

We know the integral:

∫ (1 / sin(v)) dv = ln |tan(v / 2)| + C

Thus:

ln |tan(v / 2)| = x + C

Now, substitute back ( v = y / x ):

ln |tan(y / (2 * x))| = x + C

This is the implicit solution to the differential equation.

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2. Solve: x^2 * dy + y * (x + y) * dx = 0

Solution:

We rewrite the equation as:

x^2 * dy = -y * (x + y) * dx

Divide both sides by ( x^2 * y ):

(1 / y) * dy = -(1 / x + y / x^2) * dx

Simplify:

(1 / y) * dy = -1 / x * dx - y / x^2 * dx

This equation is non-linear and may require substitution for further simplification.