Ordinary First Order Differential Equations

Question

trojanpoem · Answer

Attached:

trojanpoem · Answer

I tried:

As the function is homogeneous  then :

dy/dx = f(y/x)

y = rsintheta
x = rcostheta

y/x = tan(theta)

dy/dx = f(tan(theta))

y = xtan(theta)

dy/dx = tan(theta) + x sec^2(theta) dtheta/dx


tan(theta) + xsec^2(theta) dtheta/dx = f(tan(theta))

xdtheta/dx =  [ f((tantheta) - tan(theta) ] / sec^2 theta

sec^2 theta dtheta /  [ f((tantheta) - tan(theta) ] = dx/x



dtan(theta)/f(tantheta) - tan(theta) = ln|x| + c

trojanpoem · Answer

Which is an equation which can be solved using separation of variables.

trojanpoem · Answer

is that correct ? is there any other proof ?

jiteshmeghwal9 · Answer

according to my knowledge a homogeneous equation satisfies the condition 

$$f(ax,ay)=a^nf(x,y)$$

jiteshmeghwal9 · Answer

& we in turn can reduce the equation to be solved by variable separation method by putting y=vx

jiteshmeghwal9 · Answer

according to ur question \

y=xtan $\theta$

trojanpoem · Answer

Actually that's what the reference says:

jiteshmeghwal9 · Answer

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