Question

1. the differential equation(1+²$x$ )$dy/dx$ +$xy$ =0 , determine the particular solution when $y$ (0)=2, express y interms of $x$ .
2. show that $dy/dx=$e^(y-x) is seperable and find the general solution expessing y in terms of x
3. use the U substitution $y=tx$ to solve $(x^6-2x^5+2x^4-y^3+4x^2y)dx+(xy^2-4x^3)dy=0$

Answer 1

2.

$dy/e^y=dx/e^x$

$-e^{-y}=-e^{-x}+c$

$y=-ln(e^{-x}-c)$

1.

$dy/y=-xdx/(1+x^2)$

$lny=-ln(1+x^2)/2+lnc$

$y=c/\sqrt{1+x^2}$

$y(0)=c=2$

$y(x)=2/\sqrt{1+x^2}$

3.

$y=tx$

$(x^3-2x^2+2x-t^3+4t)dx+(t^2-4)(xdt+tdx)=0$

$(x^3-2x^2+2x)dx+(t^2-4)xdt=0$

$\int(x^2-2x+2)dx=-\int(t^2-4)dt$

$t^3/3-4t=-x^3/3+x^2-2x+c$

$y^3/(3x^3)-4y/x=-x^3/3+x^2-2x+c$