opus
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 Problem Statement
 Solve the initial value problem for the vector function ##\vec r(t)##.
 Relevant Equations

##\frac{d^2\vec r}{dt^2} = \cos(t)\hat i  12\sin(2t)\hat j  9.8\hat k## ;
##\vec r(0) = 10\hat j + 150\hat k##
##\vec r'(0) = \hat i + 6\hat j##
From ##\vec r''(t)## we integrate to get
$$\vec r'(t) = \left(\sin(t)+C_1\right)\hat i + \left(6\cos(2t)+C_2\right)\hat j  \left(9.8t+C_3\right)\hat k$$
Solving for the C constants using ##\vec r'(0) = 1\hat i + 6\hat j + 0\hat k##,
##\vec r'(0) = <C_1, C_2, C_3>##
##=<1, 6, 0>##
So we now have $$\vec r'(t) = \left(\sin(t)+1\right)\hat i + \left(6\cos(2t)+6\right)\hat j  \left(9.8t\right)\hat k$$
Using the same process one more time,
##\vec r(t) = \left(\cos(t) + t + C_4\right)\hat i + \left(3\sin(2t) + 6t + C_5\right)\hat j  \left(4.9t^2 +C_6\right)\hat k##
Solving for the constants like before, and using the initial values given,
##\vec r(0) = < C_4, C_5, C_6 >##
##= <0, 10, 150>##
And we can now state the position vector as:
$$\vec r(t) = \left(\cos(t)+t\right)\hat i + \left(3sin(2t)+6t+10\right)\hat j + \left(4.9t^2+150\right)\hat k$$
Would someone mind pointing me to where I have made a mistake? This doesn't match the given solution key which is:
##\vec r(t) = \left(\cos(t) + t1\right)\hat i + \left(3\sin(2t)+10\right)\hat j + \left(1504.9t^2\right)\hat k##
$$\vec r'(t) = \left(\sin(t)+C_1\right)\hat i + \left(6\cos(2t)+C_2\right)\hat j  \left(9.8t+C_3\right)\hat k$$
Solving for the C constants using ##\vec r'(0) = 1\hat i + 6\hat j + 0\hat k##,
##\vec r'(0) = <C_1, C_2, C_3>##
##=<1, 6, 0>##
So we now have $$\vec r'(t) = \left(\sin(t)+1\right)\hat i + \left(6\cos(2t)+6\right)\hat j  \left(9.8t\right)\hat k$$
Using the same process one more time,
##\vec r(t) = \left(\cos(t) + t + C_4\right)\hat i + \left(3\sin(2t) + 6t + C_5\right)\hat j  \left(4.9t^2 +C_6\right)\hat k##
Solving for the constants like before, and using the initial values given,
##\vec r(0) = < C_4, C_5, C_6 >##
##= <0, 10, 150>##
And we can now state the position vector as:
$$\vec r(t) = \left(\cos(t)+t\right)\hat i + \left(3sin(2t)+6t+10\right)\hat j + \left(4.9t^2+150\right)\hat k$$
Would someone mind pointing me to where I have made a mistake? This doesn't match the given solution key which is:
##\vec r(t) = \left(\cos(t) + t1\right)\hat i + \left(3\sin(2t)+10\right)\hat j + \left(1504.9t^2\right)\hat k##