Answer:
Julie
Step-by-step explanation:
Hello There =)
Commutative property is used. The commutative property <span>states that two numbers can be multiplied in either order.</span>
The answer is C because it’s the same line
Answer:
D.
Step-by-step explanation:
CPCTC which stands for "corresponding parts of congruent triangles are congruent". Since ΔMQN and ΔPQN have already been proved congruent, So every angle and sides of these two triangle corresponding with each other would be congruent.
![y'=(t+y)^2-1](https://tex.z-dn.net/?f=y%27%3D%28t%2By%29%5E2-1)
Substitute
, so that
, and
![u'=u^2-1](https://tex.z-dn.net/?f=u%27%3Du%5E2-1)
which is separable as
![\dfrac{u'}{u^2-1}=1](https://tex.z-dn.net/?f=%5Cdfrac%7Bu%27%7D%7Bu%5E2-1%7D%3D1)
Integrate both sides with respect to
. For the integral on the left, first split into partial fractions:
![\dfrac{u'}2\left(\frac1{u-1}-\frac1{u+1}\right)=1](https://tex.z-dn.net/?f=%5Cdfrac%7Bu%27%7D2%5Cleft%28%5Cfrac1%7Bu-1%7D-%5Cfrac1%7Bu%2B1%7D%5Cright%29%3D1)
![\displaystyle\int\frac{u'}2\left(\frac1{u-1}-\frac1{u+1}\right)\,\mathrm dt=\int\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7Bu%27%7D2%5Cleft%28%5Cfrac1%7Bu-1%7D-%5Cfrac1%7Bu%2B1%7D%5Cright%29%5C%2C%5Cmathrm%20dt%3D%5Cint%5Cmathrm%20dt)
![\dfrac12(\ln|u-1|-\ln|u+1|)=t+C](https://tex.z-dn.net/?f=%5Cdfrac12%28%5Cln%7Cu-1%7C-%5Cln%7Cu%2B1%7C%29%3Dt%2BC)
Solve for
:
![\dfrac12\ln\left|\dfrac{u-1}{u+1}\right|=t+C](https://tex.z-dn.net/?f=%5Cdfrac12%5Cln%5Cleft%7C%5Cdfrac%7Bu-1%7D%7Bu%2B1%7D%5Cright%7C%3Dt%2BC)
![\ln\left|1-\dfrac2{u+1}\right|=2t+C](https://tex.z-dn.net/?f=%5Cln%5Cleft%7C1-%5Cdfrac2%7Bu%2B1%7D%5Cright%7C%3D2t%2BC)
![1-\dfrac2{u+1}=e^{2t+C}=Ce^{2t}](https://tex.z-dn.net/?f=1-%5Cdfrac2%7Bu%2B1%7D%3De%5E%7B2t%2BC%7D%3DCe%5E%7B2t%7D)
![\dfrac2{u+1}=1-Ce^{2t}](https://tex.z-dn.net/?f=%5Cdfrac2%7Bu%2B1%7D%3D1-Ce%5E%7B2t%7D)
![\dfrac{u+1}2=\dfrac1{1-Ce^{2t}}](https://tex.z-dn.net/?f=%5Cdfrac%7Bu%2B1%7D2%3D%5Cdfrac1%7B1-Ce%5E%7B2t%7D%7D)
![u=\dfrac2{1-Ce^{2t}}-1](https://tex.z-dn.net/?f=u%3D%5Cdfrac2%7B1-Ce%5E%7B2t%7D%7D-1)
Replace
and solve for
:
![t+y=\dfrac2{1-Ce^{2t}}-1](https://tex.z-dn.net/?f=t%2By%3D%5Cdfrac2%7B1-Ce%5E%7B2t%7D%7D-1)
![y=\dfrac2{1-Ce^{2t}}-1-t](https://tex.z-dn.net/?f=y%3D%5Cdfrac2%7B1-Ce%5E%7B2t%7D%7D-1-t)
Now use the given initial condition to solve for
:
![y(3)=4\implies4=\dfrac2{1-Ce^6}-1-3\implies C=\dfrac3{4e^6}](https://tex.z-dn.net/?f=y%283%29%3D4%5Cimplies4%3D%5Cdfrac2%7B1-Ce%5E6%7D-1-3%5Cimplies%20C%3D%5Cdfrac3%7B4e%5E6%7D)
so that the particular solution is
![y=\dfrac2{1-\frac34e^{2t-6}}-1-t=\boxed{\dfrac8{4-3e^{2t-6}}-1-t}](https://tex.z-dn.net/?f=y%3D%5Cdfrac2%7B1-%5Cfrac34e%5E%7B2t-6%7D%7D-1-t%3D%5Cboxed%7B%5Cdfrac8%7B4-3e%5E%7B2t-6%7D%7D-1-t%7D)