Question

Help with differential Equation question.
I can do the integration factor and exact equation part but how does one use parametrisation?
Use parametrisation first, derive the equation including y and p =dy/dx
and use the ntegrating factor method to reduce it to an exact. Leave the solution in implicit parametric form.
y^3+ y^2= xy'y

Answer 1

What first came to my head when reading this question was to bring it to the form of a Bernoulli Equation, but they mention it to be exact?

Answer 2

\[\Large xyy'=y^3+y^2 \\ \Large y'=\frac{y^3+y^2}{xy}=\frac{y^2}{x}+\frac{y}{x} \\
\Large y'= \frac{1}{x}y+\frac{1}{x}y^2 \\ \Large \frac{y'}{y^2}= \frac{1}{x}\cdot \frac{1}{y}+ \frac{1}{x} \]

Answer 3

\[\Large u= \frac{1}{y} \longrightarrow u' = -\frac{y'}{y^2} \]

Answer 4

\[\Large -u'= \frac{1}{x} u+\frac{1}{x}\]

Answer 5

\[\Large u'+\frac{1}{x}u=-\frac{1}{x} \]

Answer 6

yea that seems, however I think we are suppose to parametrize it in terms of y and p first. It's pretty weird

Answer 7

thanks