Question

dy/dx= x^4y^3-x^2y
কেউ পারবেন​

Answer 1

Answer:

(2x−y+1)dx+(−x+4y−3)dy=0  

First we test for exactness for,

M(x,y)dx+N(x,y)dy=0  

∂M∂y=∂N∂x  

Here we find

∂M∂y=−1,∂N∂x=−1  

Thus it is an exact ODE. This can be solved with ease as

u(x,y)=∫Mdx+k(y), or ,u(x,y)=∫Ndy+l(x)  

Lets just use  M  for no particular reason.

u(x,y)=∫2x−y+1dx+k(y)  

u(x,y)=x2−xy+x+k(y)  

Next since we know that  ∂u∂y=N  

∂u∂y=−x+dkdy=N=−x+4y−3  

⟹dkdy=4y−3  

⟹k(y)=2y2−3y+C  

∴u(x,y)=x2−xy+x+2y2−3y=C

hope it help