Question

Determine whether the equation is exact . If it is , then solve it . (e ^ t * y + t * e ^ t * y) * d * t + (t * e ^ t + 2) * d * y = 0 .

Answer 1

$(ye^{t}+tye^{t})'_y=e^{t}+te^{t}$

$(te^t+2)'_t=e^{t}+te^{t}$

$(ye^{t}+tye^{t})'_y=(te^t+2)'_t$

So, the equation is exact .

$\frac{\partial F}{\partial t}=ye^{t}+tye^{t}$

$\frac{\partial F}{\partial y}=te^t+2$

$F=\int (ye^{t}+tye^{t})dt=ye^t+tye^t-ye^t+\phi(y)=tye^t+\phi(y)$

$\frac{\partial F}{\partial y}=te^t+\phi'(y)=te^t+2$

$\phi'(y)=te^t+2$

$\phi(y)=\int (te^t+2)dy=tye^t+2y+c$

$F=2tye^t+2y+c$