AMS 361: Applied Calculus IV Homework 1 Solved

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