Question

(xy^2+x)dx + ( yx^2+y )dy =0

Answer 1

Answer:

(y^2+1)(x^2+1)=k

Step-by-step explanation:

It looks like it is possibly separatable.

Let's try.

Factor the x out on first term and factor the y out on second term:

x(y^2+1)dx+y(x^2+1)dy=0

Subtracting y(x^2+1)dy on both sides:

x(y^2+1)dx=-y(x^2+1)dy

Divide both sides by (y^2+1)(x^2+1):

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

Integrate both sides..

Use substitution u=x^2+1 and v=y^2+1...so du= 2x dx and dv=2y dy.

.5du/u=-.5dv/v

.5ln|u|=-.5ln|v|+c

Multiply both sides by 2:

ln|u|=-ln|v|+c

Plug in our substitutions :

ln|x^2+1|=-ln|y^2+1|+c

Absolute values unnecessary since insides are positive always.... also -ln(y^2+1)=ln(1/(y^2+1))

e^ on both sides:

x^2+1=k(1/y^2+1)

(y^2+1)(x^2+1)=k