Differential equation separable or homogeneous

Today we are solving separable or homogeneous first order equation.

In the equations

A(x) + B(y) dy/dx = 0

or A(x) dx + B(y) dy = 0

we have separated the variables

Integrate both sides

( integral of A(x) dx ) + ( integral of B(y) dy ) = C

Example 1: 2x dx + ( — 1 / (y² + 1) ) dy = 0

x² — arc tan (y) = C

y = tan( x² — C)

Homogeneous first order equation:

dy/dx = F(y/x)

Let y = v x

dy/dx = v + x dv/dx = F(v)

so dx/ x + dv / ( v — F(v) ) = 0 is now separable equation

Example 2: dy/dx = — ( x² + y² ) / (2 x y) = — ( 1 + (y/x)² ) / (2 (y/x))

v = y / x , F(v) = — (1 + v²) /(2 v)

v — F(v) = v + ( 1 + v²) / (2 v) = ( 1 + 3 v²) / (2 v)

dx / x + 2 v dv / (1 + 3 v²) = 0

ln |x| + (1/3) ln (1 + 3 v²) = C

x³ ( 1 + 3 v²) = + or — e^(3C) = C1

x³ (1 + 3 y² / x² ) = C1

x³ + 3 x y² = C1

Please enjoy and review! Thank you!

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