Answered: "Find general solution of the following differential equation..."

College
Calculus and Analysis
Differential Equations
First order differential equations

UkusakazaL

Answered question

2021-02-21

Find general solution of the following differential equation

d y / d x = (2 y^{2} + x^{2} e^{(} - (y / x)^{2})) / (2 x y)

Answer & Explanation

dieseisB

Skilled2021-02-22Added 85 answers

$y = v x => y / x = v$
$d y / d x = v + x (d v) / d x$
$v + x (d v) / d x = (2 (v x)^{2} + x^{2} e^{(} - (v)^{2})) / (2 x (v x))$
$v + x (d v) / d x = (2 v^{2} x^{2} + x^{2} e^{(} - (v)^{2})) / (2 v x^{2})$
$v + x (d v) / d x = (x^{2} (2 v^{2} + e^{(} - (v)^{2}))) / (2 v x^{2})$ (take $x^{2}$ as common factor from numerator)
$v + x (d v) / d x = (2 v^{2} + e^{(} - (v)^{2})) / (2 v)$ (divide $x^{2} / x^{2} = 1$ )
Substract v from both sides and further simplify it
$v + x (d v) / d x - v = (2 v^{2} + e^{(} - (v)^{2})) / (2 v) - v$
$x (d v) / d x = (2 v^{2} e^{(} - (v)^{2}) - 2 v^{2}) / (2 v)$
$x (d v) / d x = e^{(} - (v)^{2}) / (2 x)$
$(2 v) / e^{(} - (v)^{2}) d v = d x / x$
Integrate both sides
$\int 2 v e^{v^{2}} d v = \int d x / x + c$
Substitute $v^{2} = t$
$=> 2 v d v = d t$
Hence, $\int e^{t} d t = \int d x / x + c$
$e^{t} = \ln x + c$
$t = \ln (\ln x + c)$ (use property $e^{x} = y => x = \ln y$ )
Substitute back $t = v^{2}$
$v^{2} = \ln (\ln x + c)$
Substitute back $v = y / x$
$(y / x)^{2} = \ln (\ln x + c)$
$y^{2} / x = \ln (\ln x + c)$
$y^{2} = x^{2} \ln (\ln x + c)$
$y = + - \sqrt{x^{2} \ln (\ln x + c)}$
$y = \pm x \sqrt{\ln (\ln x + C)}$

Jeffrey Jordon

Expert2021-12-12Added 2605 answers

Answer is given below (on video)

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Differential Equations

Didn't find what you were looking for?