Answer with Step-by-step explanation:
The given differential equation is
![(2x+5y)dx+(5x-4y)dy=0](https://tex.z-dn.net/?f=%282x%2B5y%29dx%2B%285x-4y%29dy%3D0)
Now the above differential equation can be re-written as
![P(x,y)dx+Q(x,y)dy=0](https://tex.z-dn.net/?f=P%28x%2Cy%29dx%2BQ%28x%2Cy%29dy%3D0)
Checking for exactness we should have
![\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20P%7D%7B%5Cpartial%20y%7D%3D%5Cfrac%7B%5Cpartial%20Q%7D%7B%5Cpartial%20x%7D)
![\frac{\partial P}{\partial y}=\frac{\partial (2x+5y)}{\partial y}=5](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20P%7D%7B%5Cpartial%20y%7D%3D%5Cfrac%7B%5Cpartial%20%282x%2B5y%29%7D%7B%5Cpartial%20y%7D%3D5)
![\frac{\partial Q}{\partial x}=\frac{\partial (5x-4y)}{\partial x}=5](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20Q%7D%7B%5Cpartial%20x%7D%3D%5Cfrac%7B%5Cpartial%20%285x-4y%29%7D%7B%5Cpartial%20x%7D%3D5)
As we see that the 2 values are equal thus we conclude that the given differential equation is exact
The solution of exact differential equation is given by
![u(x,y)=\int P(x,y)dx+\phi(y)\\\\u(x,y)=\int (2x+5y)dx+\phi (y)\\\\u(x,y)=x^2+5xy+\phi (y)](https://tex.z-dn.net/?f=u%28x%2Cy%29%3D%5Cint%20P%28x%2Cy%29dx%2B%5Cphi%28y%29%5C%5C%5C%5Cu%28x%2Cy%29%3D%5Cint%20%282x%2B5y%29dx%2B%5Cphi%20%28y%29%5C%5C%5C%5Cu%28x%2Cy%29%3Dx%5E2%2B5xy%2B%5Cphi%20%28y%29)
The value of
can be obtained by differentiating u(x,y) partially with respect to 'y' and equating the result with P(x,y)
![\frac{\partial u}{\partial y}=\frac{\partial (x^2+5xy+\phi (y)))}{\partial y}=Q(x,y))\\\\5y+\phi '(y)=(5x-4y)\\\\\phi '(y)=5x-9y\\\\\int\phi '(y)\partial y=\int (5x-9y)\partial y\\\\\phi (y)=5xy-\frac{9y^2}{2}\\\\\therefore u(x,y)=x^2+10xy-\frac{9y^2}{2}+c](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20u%7D%7B%5Cpartial%20y%7D%3D%5Cfrac%7B%5Cpartial%20%28x%5E2%2B5xy%2B%5Cphi%20%28y%29%29%29%7D%7B%5Cpartial%20y%7D%3DQ%28x%2Cy%29%29%5C%5C%5C%5C5y%2B%5Cphi%20%27%28y%29%3D%285x-4y%29%5C%5C%5C%5C%5Cphi%20%27%28y%29%3D5x-9y%5C%5C%5C%5C%5Cint%5Cphi%20%27%28y%29%5Cpartial%20y%3D%5Cint%20%285x-9y%29%5Cpartial%20y%5C%5C%5C%5C%5Cphi%20%28y%29%3D5xy-%5Cfrac%7B9y%5E2%7D%7B2%7D%5C%5C%5C%5C%5Ctherefore%20u%28x%2Cy%29%3Dx%5E2%2B10xy-%5Cfrac%7B9y%5E2%7D%7B2%7D%2Bc)
Answer:
The battle that resulted in both a Confederate victory and the death of General “Stonewall” Jackson was
Step-by-step explanation:
The battle that resulted in both a Confederate victory and the death of General “Stonewall” Jackson was
2 is distribution blah blah