AMS 361: Applied Calculus IV
Homework 1
Problem 1.1 (15 points): Verify by substitution whether the given functions are solutions of the given DE. Primes denote derivatives with respect to 𝑥.
𝑦′′ + 𝑦′ = sin 20𝑥 ; 𝑦1 = cos 𝑥 + sin 𝑥 , 𝑦2 = cos 20𝑥 + sin 𝑥 , 𝑦3 = cos𝑥 + sin 20𝑥
Problem 1.2 (15 points): Verify that satisfies the given DE and then determine a value of the constant 𝐶 so that 𝑦 satisfies the given initial condition (IC).
𝑦′ − 7𝑥6𝑦 = 0; 𝑦 , 𝑦(0) = 2020
Problem 1.3 (20 points): Find the PS of the IVP:
𝑦′ sin 𝑥 + 𝑦 cos𝑥 = 0
{
𝑦(𝜋/2) = 2020
Problem 1.4 (20 points): Solve the following IVP. 𝑥
𝑦𝜋
Problem 1.5 (30 points): Find the GS of the DE (Primes denote derivatives WRT 𝑥):
𝑦′ = (𝑥𝑦′ + 𝑦)𝑦2020 Hint: Recall relationship 𝑦′ = 𝑑𝑦 = (𝑑𝑥)−1 and regard 𝑥 as DV and 𝑦 as IV. 𝑑𝑥 𝑑𝑦
