Answer to Question #292775 in Differential Equations for luna

Question

Solve the first order linear inhomogeneous differential equation using the bernoulli method

xy^,-2y=2x⁴

Answer 1

$y' - 2\frac{y}{x} = 2{x^3}$

Substitution: $y = uv \Rightarrow y' = u'v + uv'$

Then

$u'v + uv' -2 \frac{{uv}}{x} = 2{x^3}$

$u'v + u\left( {v' - 2\frac{v}{x}} \right) = 2{x^3}$

Let

$v' - \frac{{2v}}{x} = 0 \Rightarrow \frac{{dv}}{{dx}} = \frac{{2v}}{x} \Rightarrow \frac{{dv}}{v} = \frac{{2dx}}{x} \Rightarrow \ln v = \ln {x^2} \Rightarrow v = {x^2}$

Then

$u'{x^2} = 2{x^3} \Rightarrow u' = 2x \Rightarrow u = {x^2} + C \Rightarrow y = uv = \left( {{x^2} + C} \right){x^2} = {x^4} + C{x^2}$

Answer: $y = {x^4} + C{x^2}$