Answer:
The solution for this differential equation is ![y=\sqrt{-2cos(x)+6}](https://tex.z-dn.net/?f=y%3D%5Csqrt%7B-2cos%28x%29%2B6%7D)
Step-by-step explanation:
This differential equation
is a separable First-Order ordinary differential equation.
We know this because a first-order differential equation is separable if and only if it can be written as
where <em>f</em> and <em>g</em> are known functions.
And we have
![\frac{dy}{dx}=\frac{sin(x)}{y}\\ \frac{dy}{dx}=sin(x)\frac{1}{y}](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%3D%5Cfrac%7Bsin%28x%29%7D%7By%7D%5C%5C%20%5Cfrac%7Bdy%7D%7Bdx%7D%3Dsin%28x%29%5Cfrac%7B1%7D%7By%7D)
To solve this differential equation we need to integrate both sides
![y\cdot dy=sin(x)\cdot dx\\ \int\limits {y\cdot dy}= \int\limits {sin(x)\cdot dx}](https://tex.z-dn.net/?f=y%5Ccdot%20dy%3Dsin%28x%29%5Ccdot%20dx%5C%5C%20%5Cint%5Climits%20%7By%5Ccdot%20dy%7D%3D%20%5Cint%5Climits%20%7Bsin%28x%29%5Ccdot%20dx%7D)
![\int\limits {y\cdot dy}=\frac{y^{2} }{2} + C](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7By%5Ccdot%20dy%7D%3D%5Cfrac%7By%5E%7B2%7D%20%7D%7B2%7D%20%2B%20C)
![\int\limits {sin(x) \cdot dx}=-cos(x) + C](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7Bsin%28x%29%20%5Ccdot%20dx%7D%3D-cos%28x%29%20%2B%20C)
![\frac{y^{2} }{2} + C=-cos(x) + C](https://tex.z-dn.net/?f=%5Cfrac%7By%5E%7B2%7D%20%7D%7B2%7D%20%2B%20C%3D-cos%28x%29%20%2B%20C)
We can make a new constant of integration ![C_{1}](https://tex.z-dn.net/?f=C_%7B1%7D)
![\frac{y^{2} }{2}=-cos(x) + C_{1}](https://tex.z-dn.net/?f=%5Cfrac%7By%5E%7B2%7D%20%7D%7B2%7D%3D-cos%28x%29%20%2B%20C_%7B1%7D)
We need to isolate y
![\frac{y^{2} }{2}=-cos(x) + C_{1}\\y^2=-2cos(x)+2*C_{1}\\\mathrm{For\:}y^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}y=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}\\y=\sqrt{-2cos(x)+c_{1} } \\y=-\sqrt{-2cos(x)+c_{1} }](https://tex.z-dn.net/?f=%5Cfrac%7By%5E%7B2%7D%20%7D%7B2%7D%3D-cos%28x%29%20%2B%20C_%7B1%7D%5C%5Cy%5E2%3D-2cos%28x%29%2B2%2AC_%7B1%7D%5C%5C%5Cmathrm%7BFor%5C%3A%7Dy%5E2%3Df%5Cleft%28a%5Cright%29%5Cmathrm%7B%5C%3Athe%5C%3Asolutions%5C%3Aare%5C%3A%7Dy%3D%5Csqrt%7Bf%5Cleft%28a%5Cright%29%7D%2C%5C%3A%5C%3A-%5Csqrt%7Bf%5Cleft%28a%5Cright%29%7D%5C%5Cy%3D%5Csqrt%7B-2cos%28x%29%2Bc_%7B1%7D%20%7D%20%5C%5Cy%3D-%5Csqrt%7B-2cos%28x%29%2Bc_%7B1%7D%20%7D)
We have the initial conditions y(0)=2 so we can find the value of the constant of integration for ![y=\sqrt{-2cos(x)+c_{1} }](https://tex.z-dn.net/?f=y%3D%5Csqrt%7B-2cos%28x%29%2Bc_%7B1%7D%20%7D%20)
![2=\sqrt{-2\cos \left(0\right)+c_1}\\2= \sqrt{-2+c_1} \\c_1=6](https://tex.z-dn.net/?f=2%3D%5Csqrt%7B-2%5Ccos%20%5Cleft%280%5Cright%29%2Bc_1%7D%5C%5C2%3D%20%5Csqrt%7B-2%2Bc_1%7D%20%5C%5Cc_1%3D6)
For
there is not solution for
in the domain of real numbers.
The solution for this differential equation is ![y=\sqrt{-2cos(x)+6}](https://tex.z-dn.net/?f=y%3D%5Csqrt%7B-2cos%28x%29%2B6%7D)
All we have to do is divide 192 by 8
192 ÷ 8
![\frac{192}{8}](https://tex.z-dn.net/?f=%20%5Cfrac%7B192%7D%7B8%7D%20)
= 24
there were
24 packs of frozen peas in each box
Vertex is at (4,−6)and the axis of symmetry is x=4
Before you write three numbers between 0.33 and 0.34 , you need to indicate whether you want rational numbers or real numbers (that is including irrational numbers).
You can factor a parabola by finding its roots: if
![p(x)=x^2+bx+c](https://tex.z-dn.net/?f=%20p%28x%29%3Dx%5E2%2Bbx%2Bc%20)
has roots
, then you have the following factorization:
![p(x) = (x-x_1)(x-x_2)](https://tex.z-dn.net/?f=%20p%28x%29%20%3D%20%28x-x_1%29%28x-x_2%29%20)
In order to find the roots, you can use the usual formula
![x_{1,2} = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=%20x_%7B1%2C2%7D%20%3D%20%5Cdfrac%7B-b%5Cpm%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%20)
In the first example, this formula leads to
![x_{1,2} = \dfrac{-2\pm\sqrt{4+4}}{2} = \dfrac{-2\pm\sqrt{8}}{2} = \dfrac{-2\pm2\sqrt{2}}{2} = 1 \pm \sqrt{2}](https://tex.z-dn.net/?f=%20x_%7B1%2C2%7D%20%3D%20%5Cdfrac%7B-2%5Cpm%5Csqrt%7B4%2B4%7D%7D%7B2%7D%20%3D%20%5Cdfrac%7B-2%5Cpm%5Csqrt%7B8%7D%7D%7B2%7D%20%3D%20%5Cdfrac%7B-2%5Cpm2%5Csqrt%7B2%7D%7D%7B2%7D%20%3D%201%20%5Cpm%20%5Csqrt%7B2%7D%20)
So, you can factor
![x^2-2x-1 = (x-1-\sqrt{2})(x-1+\sqrt{2})](https://tex.z-dn.net/?f=%20x%5E2-2x-1%20%3D%20%28x-1-%5Csqrt%7B2%7D%29%28x-1%2B%5Csqrt%7B2%7D%29%20)
The same goes for the second parabola.
As for the third exercise, simply plug the values asking
![f(1.5)=-5.25](https://tex.z-dn.net/?f=%20f%281.5%29%3D-5.25%20)
you get
![f(-1.5) = 1.5c-3 = -5.25](https://tex.z-dn.net/?f=%20f%28-1.5%29%20%3D%201.5c-3%20%3D%20-5.25%20)
Add 3 to both sides:
![1.5c = -2.25](https://tex.z-dn.net/?f=%201.5c%20%3D%20-2.25%20)
Divide both sides by 1.5:
![c = 1.5](https://tex.z-dn.net/?f=%20c%20%3D%201.5%20)