austinL
  • austinL
Find the solution of the given initial value problem. \(ty\prime +2y=t^2-t+1\) \(y(1)=0\) I have a rough idea of how to go about this, but I am still feeling out how is best to start about solving each type of problem.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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austinL
  • austinL
Ok, so i need to get it in the standard for yeah?
austinL
  • austinL
\(y\prime+\frac{2}{t}y=t-1+\frac{1}{t}\) That look ok?
anonymous
  • anonymous
multiply both sides by t and you will get \[\frac{ d }{ dt }\left( t ^{2} y\right)\] on the left side

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anonymous
  • anonymous
and then seperation of variables
anonymous
  • anonymous
first solve hom. part\[\Large ty'+2y=0 \]this is separable..
anonymous
  • anonymous
for particular sol. search by assuming y(t)= At^2+Bt+C is sol..
anonymous
  • anonymous
\[\Large ty'+2y=0\\ \Large ty'=-2y\\ \Large\frac{dy}{y}=\frac{-2}{t}dt \]
anonymous
  • anonymous
if you multiply both sides by t \[t ^{2}\frac{ dy }{ dt }+2ty=t ^{3}-t ^{2}+t\]\[t ^{2}\frac{ dy }{ dt }+\frac{ d \left( t ^{2} \right) }{ dt }y=t ^{3}-t ^{2}+t\]\[\frac{ d \left( t ^{2}y \right) }{ dt }=t ^{3}-t ^{2}+t\]\[d \left( t ^{2}y \right)=\left( t ^{3}-t ^{2}+t \right)dt\]
anonymous
  • anonymous
now integrate both sides and you will get y(t)

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