GATE 2024 IN – Question 64
The differential equation $y^{\prime\prime\prime}+3y^{\prime\prime}+3y^\prime+y=30e^{-t}$ has solution $y(t)=(c_0+c_1t-c_2t^2+c_3t^3)e^{-t}$. Given $y(0)=3$, $y^\prime(0)=-3$, $y^{\prime\prime}(0)=-47$, the value of $c_0+c_1+c_2+c_3$ is ______ (nearest integer).
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Correct answer: 33 to 33
Explanation
Write $y=e^{-t}P(t)$. Then $(D+1)^3y=e^{-t}P^{\prime\prime\prime}=30e^{-t}$, so $6c_3=30$ and $c_3=5$. Initial conditions yield $c_0=3$, $c_1-c_0=-3$ so $c_1=0$, and $-2c_2-2c_1+c_0=-47$ so $c_2=25$. Their sum is $3+0+25+5=\boxed{33}$.