Vtyjkyuk

The solution of $dfracdydx= dfrac(x+y)^2(x+2)(y-2)$ is given by:

a) $(x+2)^4 (1+frac2yx)= ke^frac2yx$

b) $(x+2)^4 (1+ 2frac(y-2)x+2)= ke^frac2(y-2)x+2$

c) $(x+2)^3 (1+ 2frac(y-2)x+2)= ke^frac2(y-2)x+2$

d) None of these

Attempt:

I have expanded and checked but couldn't spot any exact differentials.

Secondly, it's not a homogeneous equation, so couldn't use $y = vx$.

How do I go about solving this problem?

hint: $x+y = (x+2) + (y-2)$ , and use $(A+B)^2 = A^2 + 2AB +B^2, A = x+2,B = y - 2$ to expand the numerator and simplify

If there is no typo in the equation, I do not think that a solution could be obtained.

The best I was able to do was, working with $x(y)$ instead of $y(x)$
$$x'=frac(y-2) (x+2)(x+y)^2$$ Making $x=z-2$ gives
$$z'=frac(y-2) z(z+y-2)^2 $$ which leads to an implicit equation
$$fracz (3 log (z)+log (z+2 y-4)-1)-2 y+44 z=C$$ which looks impossible to solve.

Let $X=x+2$ and $Y=y-2$, so the given DE is equivalent to
$$dfracmathrm d Ymathrm d X=frac(X+Y)^2X Y$$
last DE is homogeneous, so can be transformed into a separable DE by making $Y =uX$ as follows
$$dfracmathrm d Ymathrm d X=frac(X+Y)^2X Yqquadiffqquad u+Xfracmathrm d umathrm dX=fracX^2(1+u)^2X^2u$$
Then we get
$$Xfracmathrm d umathrm d X=frac1+2uuquadimpliesquad fracu2u+1dfracmathrm d umathrm d X=frac1X$$
Integrating both sides respect to $X$ (supposing that $u$ depends on $X$) we get
beginalign*
intfracu2u+1dfracmathrm d umathrm d Xmathrm d X&=intfrac1Xmathrm dX\[4pt]
intleft(frac12-frac1/22u+1right)mathrm du&=intfrac1Xmathrm dX\[4pt]
frac12u-frac14ln|2u+1|&=ln|X|+c_1
endalign*
Last equality is equivalent to
$$fracy-2x+2-frac12lnleft|frac2(y-2)x+2+1right|=2ln|x+2|+2c_1$$
Notice that this solution can be carried to the form b):
beginalign*
frac2(y-2)x+2-4c_2&=4ln|x+2|+lnleft|frac2(y-2)x+2+1right|\[4pt]
k e^frac2(y-2)x+2&=(x+4)^4left[frac2(y-2)x+2+1right]
endalign*
where $k=pm e^-4c_1$

$$dfracdydx= dfrac(x+y)^2(x+2)(y-2)=left(dfracx+yx+2right)^2dfracx+2y-2$$
let $w=dfracx+yx+2$ then $y=w(x+2)-x$ and
$$w'(x+2)+w-1=y'=w^2dfrac1w-1$$
which is separable
$$dfracw-12w-1dw=dfracdxx+2$$
with integration the solution will be obtained.