Solving for Particular Solution in a Differential Equation

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To find the particular solution for the differential equation 2x'' + x = 3t^2, the user proposes a solution of the form At^2 + Bt + C. By differentiating and substituting back into the equation, they derive the expression At^2 + (4A + B)t + (2B + C) = 3t^2. The user seeks confirmation on the correctness of their values for A, B, and C, which they calculated as A=3, B=-12, and C=24. The discussion focuses on verifying the accuracy of these coefficients in solving the differential equation.
rbailey5
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Homework Statement


determine the particular solution for the differential equation 2x^double prime+x=3t^2


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The Attempt at a Solution


since F(t)=3t^2 I used At^2+Bt+C and the first derivative is 2A+B

plugging back in I get
At^2+(4A+B)t+(2B+C)=3t^2

is this correct?
how do I solve for the variables?
 
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I got A=3, B=-12, and C=24 does that look right?
 

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