Lecture 6 Notes

that is find the y 2y 3y e2t step 1 solve

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Unformatted text preview: cients (3.5) • Example 4. Define the operator general solution to L[y ] = e 2t L[y ] = y + 2y − 3y. ￿￿ . That is, ￿ Find the y ￿￿ + 2y ￿ − 3y = e2t . • Step 1: Solve the associated homogeneous equation y ￿￿ + 2y ￿ − 3y = 0. Method of undetermined coefficients (3.5) • Example 4. Define the operator general solution to L[y ] = e 2t L[y ] = y + 2y − 3y. ￿￿ . That is, ￿ y ￿￿ + 2y ￿ − 3y = e2t . • Step 1: Solve the associated homogeneous equation y ￿￿ + 2y ￿ − 3y = 0. yh (t) = C1 e + C2 e t Find the −3t Method of undetermined coefficients (3.5) • Example 4. Define the operator general solution to L[y ] = e 2t L[y ] = y + 2y − 3y. ￿￿ . That is, ￿ Find the y ￿￿ + 2y ￿ − 3y = e2t . • Step 1: Solve the associated homogeneous equation y ￿￿ + 2y ￿ − 3y = 0. yh (t) = C1 e + C2 e t −3t • Step 2: What do you have to plug in to L[ 2t · ] to get e out? Method of undetermined coefficients (3.5) • Example 4. Define the operator general solution to L[y ] = e 2t L[y ] = y + 2y − 3y. ￿￿ . That is, ￿ Find the y ￿￿ + 2y ￿ − 3y = e2t . • Step 1: Solve the associated homogeneous equation y ￿￿ + 2y ￿ − 3y = 0. yh (t) = C1 e + C2 e t −3t • Step 2: What do you have to plug in to L[ • Try yp (t) = Ae . 2t 2t · ] to get e out? Method of undetermined coefficients (3.5) • Example 4. Define the operator general solution to L[y ] = e 2t L[y ] = y + 2y − 3y. ￿￿ . That is, ￿ Find the y ￿￿ + 2y ￿ − 3y = e2t . • Step 1: Solve the associated homogeneous equation y ￿￿ + 2y ￿ − 3y = 0. yh (t) = C1 e + C2 e t −3t • Step 2: What do you have to plug in to L[ • Try yp (t) = Ae . • L[yp (t)] 2t = L[Ae2t ] = 2t · ] to get e out? Method of undetermined coefficients (3.5) • Example 4. Define the operator general solution to L[y ] = e 2t L[y ] = y + 2y − 3y. ￿￿ . That is, ￿ Find the y ￿￿ + 2y ￿ − 3y = e2t . • Step 1: Solve the associated homogeneous equation y ￿￿ + 2y ￿ − 3y = 0. yh (t) = C1 e + C2 e −3t t 2t • Step 2: What do you have to plug in to L[ • Try yp (t) = Ae . • L[yp (t)] 2t = L[Ae2t ] = · ] to get e out? (C) (A) 5e (B) 5Ae 2t 2t 4e (D) 4Ae 2t 2t Method of undetermined coefficients (3.5) • Example 4. Define the operator general solution to L[y ] = e 2t L[y ] = y + 2y − 3y. ￿￿ . That is, ￿ Find the y ￿￿ + 2y ￿ − 3y = e2t . • Step 1: Solve the associated homogeneous equation y ￿￿ + 2y ￿ − 3y = 0. yh (t) = C1 e + C2 e −3t t 2t • Step 2: What do you have to plug in to L[ • Try yp (t) = Ae . • L[yp (t)] 2t = L[Ae2t ] = · ] to get e out? (C) (A) 5e (B) 5Ae 2t 2t 4e (D) 4Ae 2t 2t Method of undetermined coefficients (3.5) • Example 4. Define the operator general solution to L[y ] = e 2t L[y ] = y + 2y − 3y. ￿￿ . That is, ￿ Find the y ￿￿ + 2y ￿ − 3y = e2t . • Step 1: Solve the associated homogeneous equation y ￿￿ + 2y ￿ − 3y = 0. yh (t) = C1 e + C2 e −3t t 2t • Step 2: What do you have to plug in to L[ • Try yp (t) = Ae . • L[yp (t)] 2t = L[Ae2t ] = · ] to get e out? (C) (A) 5e (B) 5Ae 2t 2t 4e (D) 4Ae 2t 2t • A is an undetermined coefficient (until you determine it). Method of undetermined coefficients (3.5) • Example 4. Define the operator general solution to L[y ] = e 2t L[C1 e + C2 e t L[y ] = y + 2y − 3y. ￿￿ . That is, −3t ￿ Find the y ￿￿ + 2y ￿ − 3y = e2t . + Ae ] = 0 + 5Ae 2t 2t Method of undetermined coefficients (3.5) • Example 4. Define the operator general solution to L[y ] = y + 2y − 3y. ￿￿ L[y ] = e 2t . That is, • Summarizing: L[C1 e + C2 e t −3t ￿ Find the y ￿￿ + 2y ￿ − 3y = e2t . + Ae ] = 0 + 5Ae 2t 2t Method of undetermined coefficients (3.5) • Example 4. Define the operato...
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