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)
Answers:
tan(A) = 8/15
sin(A) = 8/17
cos(A) = 15/17
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Explanation:
The tangent ratio involves the opposite over adjacent. With respect to angle A, the opposite side is 8 units long as this side is as far as possible away from angle A. In contrast, the adjacent side is 15 units long because this leg is closest possible and the two are right next to one another.
Therefore,
tan(angle) = opposite/adjacent
tan(A) = BC/AC
tan(A) = 8/15
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As for the other trig ratios, they are:
sin(angle) = opposite/hypotenuse
cos(angle) = adjacent/hypotenuse
which explains why sin(A) and cos(A) are 8/17 and 15/17 respectively.
Answer:
{xer}
Step-by-step explanation:
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<span>In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/ x or x −1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a / b is b / a. For the multiplicative inverse of a real number, divide 1 by the number.</span>
Answer:
x = 6
Step-by-step explanation:
we both know you dont wanna know how you just want the answer