Which of the following is true for y(x) that satisfies the differential equation [dy dx] =…

Question

Which of the following is true for y(x) that satisfies the differential equation

$${{dy} \over {dx}}$$ = xy - 1 + x - y; y(0) = 0 :

Accepted Answer

$${{dy} \over {dx}} = (x - 1)y + (x - 1)$$

$${{dy} \over {dx}} = (x - 1)(y + 1)$$

$${{dy} \over {y + 1}} = (x - 1)dx$$

Integrating both sides, we get

$$\ln (y + 1) = {{{x^2}} \over 2} - x + c$$

$x = 0,y = 0$

$$ \Rightarrow c = 0$$

$$ \therefore $$ $$\ln (y + 1) = {{{x^2}} \over 2} - x$$

putting $$x = 1,\ln (y + 1) = {1 \over 2} - 1 =  - {1 \over 2}$$

$$y + 1 = {e^{ - {1 \over 2}}}$$

$$y = {e^{ - {1 \over 2}}} - 1$$

$$ \therefore $$ $$y(1) = {e^{ - {1 \over 2}}} - 1$$

Solutions · 1