Question

\[(2t^3y-y^3)\text dt-2t^4\text dy=0\]

Answer 1

\[\frac{\partial M}{\partial y}=2t^3-3y^2\]\[\frac{\partial N}{\partial t}=-8t^3\]\[\frac{\partial M}{\partial y}\neq\frac{\partial N}{\partial t}\]
\[R=R(t,y)=t^my^n\]\[\frac{\partial MR(t,y)}{\partial y}=\frac{\partial NR(t,y)}{\partial t}\]\[R\frac{\partial M}{\partial y}+R_yM=R\frac{\partial N}{\partial t}+R_tN\]
\[t^my^n(2t^3)+nt^my^{n-1}(2t^3y)=t^my^n(-8t^3)+mt^{m-1}y^n(-2t^4)\]\[2t^{m+3}y^n+2nt^{m+3}y^{n}=-8t^{m+3}y^n-2mt^{m+3}y^n\]\[(10+2n+2m)t^{m+3}y^n=0\]\[(10+2n+2m)=0\]

Answer 2

how can i find \(n ~~\& ~~m\)

Answer 3

i need two equations to solve two variables right?

Answer 4

\[10+2n+2m=0\]
\[2n=-2m-10\]
\[-2n=2m+10\]
\[n=-m-5\]
\[-2(-m-5)=2m+10 \]
\[2m+10=2m+10\]
\[0=0\]
like i said Wislar, i need two equations , not just two versions of the same equation

Answer 5

\[m+3=0\]?

Answer 6

How m+ 3 = 0 ??

Answer 7

i dont know, i copied it from the black board

Answer 8

\[(10+2n+2m)t^{m+3}y^n=0\]

Answer 9

By the way I know none of it..

In which topic is this ??

Differential or Partial Differential Equations ??

Answer 10

first order exact equations with an integrating factor a function of both variables