First set is parallel - same gradient
Second set is neither - gradient is 2/6 or 1/3. It then had to be 3 for it to be perpendicular.
Hope this helps!
Answer:
satisfies given homogenous solution and the particular solution is
.
Step-by-step explanation:
Given homogeneous equation is,
where t>0 subject to ![y_1(t)=t^2, y_2(t)=\frac{1}{t}\hfill (1)](https://tex.z-dn.net/?f=y_1%28t%29%3Dt%5E2%2C%20y_2%28t%29%3D%5Cfrac%7B1%7D%7Bt%7D%5Chfill%20%281%29)
To verify,
- wheather
satisfies given homogenous equation, then both will satisfy
that is,
![t^2y_{1}^{''}-2y_1=t^2\times 2-2\times t^2=0](https://tex.z-dn.net/?f=t%5E2y_%7B1%7D%5E%7B%27%27%7D-2y_1%3Dt%5E2%5Ctimes%202-2%5Ctimes%20t%5E2%3D0)
![t^2y_{2}^{''}-2y_2=t^2\times 2t^{-3}-2\times t^{-1}=2t^{-1}-2t^{-1}=0](https://tex.z-dn.net/?f=t%5E2y_%7B2%7D%5E%7B%27%27%7D-2y_2%3Dt%5E2%5Ctimes%202t%5E%7B-3%7D-2%5Ctimes%20t%5E%7B-1%7D%3D2t%5E%7B-1%7D-2t%5E%7B-1%7D%3D0)
Thus
and
satisfies (1).
Now the wronskean,
![W(y_1, y_2)(x)=y_1y_{2}^{'}-y_{2}y_{1}^{'}=-t^2t^{-2}-2tt^{-1}=-3\neq 0](https://tex.z-dn.net/?f=W%28y_1%2C%20y_2%29%28x%29%3Dy_1y_%7B2%7D%5E%7B%27%7D-y_%7B2%7Dy_%7B1%7D%5E%7B%27%7D%3D-t%5E2t%5E%7B-2%7D-2tt%5E%7B-1%7D%3D-3%5Cneq%200)
Thus
and
are solution of (1) .
where ![D\equiv \frac{\partial }{\partial t}](https://tex.z-dn.net/?f=D%5Cequiv%20%5Cfrac%7B%5Cpartial%20%7D%7B%5Cpartial%20t%7D)
![=-\frac{1}{2t^2}\frac{1}{1-\frac{D^2}{t^2}}()5t^2-1](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B1%7D%7B2t%5E2%7D%5Cfrac%7B1%7D%7B1-%5Cfrac%7BD%5E2%7D%7Bt%5E2%7D%7D%28%295t%5E2-1)
![=-\frac{1}{2t^2}(1-\frac{D^2}{t^2})^{-1}(5t^2-1)](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B1%7D%7B2t%5E2%7D%281-%5Cfrac%7BD%5E2%7D%7Bt%5E2%7D%29%5E%7B-1%7D%285t%5E2-1%29)
![=-\frac{1}{2t^2}(1-\frac{D^2}{t^2}+........)(5t^2-1)](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B1%7D%7B2t%5E2%7D%281-%5Cfrac%7BD%5E2%7D%7Bt%5E2%7D%2B........%29%285t%5E2-1%29)
sincce ![D^2(5t^2-1)=10](https://tex.z-dn.net/?f=D%5E2%285t%5E2-1%29%3D10)
![=\frac{5}{t^4}+\frac{1}{2t^2}-\frac{5}{2}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B5%7D%7Bt%5E4%7D%2B%5Cfrac%7B1%7D%7B2t%5E2%7D-%5Cfrac%7B5%7D%7B2%7D)
Hence the result.
Answer:
Step-by-step explanation:
the first one your missing 3cm and the second one your missing 1cm.
Answer:
-2
Step-by-step explanation:
You multiply each term by -2 to get the next term.
To find this for this sequence, divide the term on the right by the term on the left.
8 / -4 = -2 or -4 * -2 = 8
-16 / 8 = -2 or 8 * -2 = -16
32 / -16 = -2 or -16 * -2 = 32
-64 / 32 = -2 or 32 * -2 = -64