Problem 1.1 (15 points): Verify by substitution whether the given functions are solutions of the given DE. Primes denote derivatives with respect to π₯.
π¦β²β² + π¦β² = sin 20π₯ ;Β π¦1 = cos π₯ + sin π₯ , π¦2 = cos 20π₯ + sin π₯ , π¦3 = cosπ₯ + sin 20π₯
Problem 1.2 (15 points): Verify that Β satisfies the given DE and then determine a value of the constant πΆ so that π¦ satisfies the given initial condition (IC).
π¦β² β 7π₯6π¦ = 0;Β Β Β Β π¦ ,Β Β Β Β Β Β Β Β Β π¦(0) = 2020
Problem 1.3 (20 points): Find the PS of the IVP:
π¦β² sin π₯ + π¦ cosπ₯ = 0
{
π¦(π/2) = 2020
Problem 1.4 (20 points): Solve the following IVP. π₯
π¦π
Problem 1.5 (30 points): Find the GS of the DE (Primes denote derivatives WRT π₯):
π¦β² = (π₯π¦β² + π¦)π¦2020 Hint: Recall relationship π¦β² = ππ¦Β = (ππ₯)β1 and regard π₯ as DV and π¦ as IV. ππ₯Β Β Β Β Β Β Β ππ¦
