Question

Homogenous Equation (convert to separable eqns: v= y/x) Find the general solution, if possible otherwise find a relation that defines the solution implicitly. x^2y'-xy= (x^2+y^2)tan^-1 y/x

Help please =(

Answer 1

\[x^2y'-xy= (x^2+y^2)\tan^{-1} \frac yx\]
\[x^2y'= (x^2+y^2)\tan^{-1} \frac yx+xy\]
\[y'= \frac{(x^2+y^2)}{x^2}\tan^{-1} \frac yx+\frac yx\]
\[y'= \tan^{-1} \frac yx+\left(\frac{y}{x}\right)^2\tan^{-1} \frac yx+\frac yx\]

Answer 2

\[\frac yx=v\]
\[y=vx\]
\[y'=v+v'x\]

Answer 3

\[v+v'x= \tan^{-1} v+v^2\tan^{-1} v+v\]

Answer 4

but how do you know its homogenous equation and not exact? how do you tell the difference because on the test his gonna give me like this type and i need to know which one works and which one does not.

Answer 5

it said homogenous in the question

Answer 6

\[v'x=(1+v^2)\tan^{-1} v\]
\[\frac{\text dv}{\text dx}=\frac{(1+v^2)\tan^{-1} v}x\]
\[\frac{\text dv}{(1+v^2)\tan^{-1} v}=\frac{\text dx}x\]

Answer 7

im justing saying on the test he would not say it but this would be just one of the option but i want to know how to determine which to use

Answer 8

for a first order DE

try in this order
variables separable,
linear,
homogenous,
homogenous with a linear shift
bernoulli,
exact,
exact with integrating factor

Answer 9

if you have an exact equation on a test it will probably be written
\[M\text dx+N\text dy=0\] or\[M\text dx=-N\text dy\]

Answer 10

okay thank you so much i will ask more question if i still have any thanks for the breakdown of the order =)

Answer 11

i have all the Solutions to sets 2A,2B,2C,2D,2E,2F
if you want them i can up load them