4,5*(-3)-2,6 = -13,5-2,6 = -16,1
Умножая левую и правую части на интегрирующий множитель
, мы получим
![(x+2y)dy-dx=0~~~~|\cdot e^{-y}\\ (x+2y)e^{-y}dy-e^{-y}dx=0](https://tex.z-dn.net/?f=%28x%2B2y%29dy-dx%3D0~~~~%7C%5Ccdot+e%5E%7B-y%7D%5C%5C+%28x%2B2y%29e%5E%7B-y%7Ddy-e%5E%7B-y%7Ddx%3D0)
Дифференциальное уравнение является уравнением в полных дифференциалах, поскольку соответствующие частные производные равны:
![\displaystyle \frac{\partial Q}{\partial x}=\frac{\partial}{\partial x}(x+2y)=e^{-y};~~~~\frac{\partial P}{\partial y}=\frac{\partial}{\partial y}\left(-e^{-y}\right)=e^{-y}](https://tex.z-dn.net/?f=%5Cdisplaystyle+%5Cfrac%7B%5Cpartial+Q%7D%7B%5Cpartial+x%7D%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+x%7D%28x%2B2y%29%3De%5E%7B-y%7D%3B~~~~%5Cfrac%7B%5Cpartial+P%7D%7B%5Cpartial+y%7D%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial+y%7D%5Cleft%28-e%5E%7B-y%7D%5Cright%29%3De%5E%7B-y%7D)
![\displaystyle \left \{ {{\frac{\partial u}{\partial x}=P(x,y)} \atop {\frac{\partial u}{\partial y}=Q(x,y)}} \right.~~~\Rightarrow~~~\left \{ {{\frac{\partial u}{\partial x}=-e^{-y}} \atop {\frac{\partial u}{\partial y}}=(x+2y)e^{-y}} \right. ~~~~\Rightarrow~~~\left \{ {{u=-xe^{-y}+\phi(y)} \atop {\frac{\partial u}{\partial y}=(x+2y)e^{-y}}} \right.\\ \\ (-xe^{-y}+\phi(y))'_y=(x+2y)e^{-y}\\ \\ xe^{-y}+\phi'(y)=xe^{-y}+2ye^{-y}\\ \\ \phi'(y)=2ye^{-y}~~~\Rightarrow~~~ \phi(y)=\int2ye^{-y}dy=](https://tex.z-dn.net/?f=%5Cdisplaystyle+%5Cleft+%5C%7B+%7B%7B%5Cfrac%7B%5Cpartial+u%7D%7B%5Cpartial+x%7D%3DP%28x%2Cy%29%7D+%5Catop+%7B%5Cfrac%7B%5Cpartial+u%7D%7B%5Cpartial+y%7D%3DQ%28x%2Cy%29%7D%7D+%5Cright.~~~%5CRightarrow~~~%5Cleft+%5C%7B+%7B%7B%5Cfrac%7B%5Cpartial+u%7D%7B%5Cpartial+x%7D%3D-e%5E%7B-y%7D%7D+%5Catop+%7B%5Cfrac%7B%5Cpartial+u%7D%7B%5Cpartial+y%7D%7D%3D%28x%2B2y%29e%5E%7B-y%7D%7D+%5Cright.+~~~~%5CRightarrow~~~%5Cleft+%5C%7B+%7B%7Bu%3D-xe%5E%7B-y%7D%2B%5Cphi%28y%29%7D+%5Catop+%7B%5Cfrac%7B%5Cpartial+u%7D%7B%5Cpartial+y%7D%3D%28x%2B2y%29e%5E%7B-y%7D%7D%7D+%5Cright.%5C%5C+%5C%5C+%28-xe%5E%7B-y%7D%2B%5Cphi%28y%29%29%27_y%3D%28x%2B2y%29e%5E%7B-y%7D%5C%5C+%5C%5C+xe%5E%7B-y%7D%2B%5Cphi%27%28y%29%3Dxe%5E%7B-y%7D%2B2ye%5E%7B-y%7D%5C%5C+%5C%5C+%5Cphi%27%28y%29%3D2ye%5E%7B-y%7D~~~%5CRightarrow~~~+%5Cphi%28y%29%3D%5Cint2ye%5E%7B-y%7Ddy%3D)
![=\displaystyle \left\{\begin{array}{ccc}u=2y;~~du=2dy\\ e^{-y}dy=dv;~~v=-e^{-y}\end{array}\right\}=-2ye^{-y}+2\int e^{-y}dy=-2ye^{-y}-2e^{-y}](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle+%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bccc%7Du%3D2y%3B~~du%3D2dy%5C%5C+e%5E%7B-y%7Ddy%3Ddv%3B~~v%3D-e%5E%7B-y%7D%5Cend%7Barray%7D%5Cright%5C%7D%3D-2ye%5E%7B-y%7D%2B2%5Cint+e%5E%7B-y%7Ddy%3D-2ye%5E%7B-y%7D-2e%5E%7B-y%7D)
Общий интеграл:
![-xe^{-y}-2ye^{-y}-2e^{-y}=C\\ \\ \boxed{-(2y+x+2)e^{-y}=C}](https://tex.z-dn.net/?f=-xe%5E%7B-y%7D-2ye%5E%7B-y%7D-2e%5E%7B-y%7D%3DC%5C%5C+%5C%5C+%5Cboxed%7B-%282y%2Bx%2B2%29e%5E%7B-y%7D%3DC%7D)
График гиперболы имеет вид:
![y = \frac{k}{x} +b](https://tex.z-dn.net/?f=y+%3D++%5Cfrac%7Bk%7D%7Bx%7D+%2Bb)
Поэтому:
<span>y=−7x - прямая
</span>
![y=7x^2](https://tex.z-dn.net/?f=y%3D7x%5E2)
- парабола
<span>y=−x+17 - прямая
</span><span>y=-1x - прямая или если запись другая: </span>
![y=- \frac{1}{x}](https://tex.z-dn.net/?f=y%3D-+%5Cfrac%7B1%7D%7Bx%7D+)
- гирербола
Y = 3x^3 - 4.5x^3 = -1,5x^3
y ' = 3*(-1.5)x^2 = -4,5 x^2