Question

e ^ (- x) * d * y - (e ^ (- x) * y + y ^ 2) * d * x = 0 .

Answer 1

$$e^{-x}dy-(e^{-x}y+y^2)dx=0$$

$$y'-y=e^xy^2$$

$$u=y^{1-n}=y^{1-2}=y^{-1}$$

$$u'=-y^{-2}y'$$

$$-u'-u=e^x$$

$$\mu(x)=e^x$$

$$e^xu'+e^xu=-e^{2x}$$

$$(e^xu)'=-e^{2x}$$

Integrate

$$e^xu=\int(-e^{2x})dx$$

$$e^xu=-\dfrac{1}{2}e^{2x}+C_1$$

$$u=-\dfrac{1}{2}e^{x}+C_1e^{-x}$$

$$y(x)=\dfrac{1}{-\dfrac{1}{2}e^{x}+C_1e^{-x}}$$

$$y(x)=\dfrac{2e^x}{C-e^{2x}}$$