Spring 2024
MECH215 Engineering Math I
Homework #1 due at 3:00 pm on March 28th
(submit to the Blackboard as a single pdf file)
1. (10 points) Suppose that we consider a falling object of mass m =10 kg and drag coefficient
γ = 2 kg/sec. Then the equation of motion becomes
𝑣
𝑑𝑣
9.8
5
𝑑𝑡
Suppose this object is dropped (i.e., initial velocity is zero) from the height of
300 m. Find its velocity at any time t. How long will it take to fall to the ground,
and how fast will it be moving at the time of impact?
2. (10 points) Solve the differential equation
𝑑𝑦
2y 4 𝑡
𝑑𝑡
and discuss the behavior of solutions as 𝑡 → ∞. (Hint: you will have different
behaviors depending on the constant produced by an integral)
3. (10 points) Find the solution of the given initial value problem.
𝑥
𝑑𝑦
𝑑𝑥
2y
𝜋
2
sin𝑥 ; 𝑦
1, 𝑡
0
4. (10 points) Solve the differential equation
𝑦 cos 𝑥
2𝑥𝑒
sin 𝑥
𝑥 𝑒
1
𝑑𝑦
𝑑𝑥
0
5. (10 points) Consider the initial value problem
2𝑦′′
3𝑦′
2𝑦
0; 𝑦 0
1, 𝑦 0
𝛽
where 𝛽 0. Solve the initial value problem. Find the smallest value of 𝛽 for
which the solution has no minimum point.
6. (10 points) If the roots of the characteristic equation are real (two distinct roots or repeated),
𝑏𝑦
𝑐𝑦 0 can take on the value zero (i.e., y(x)
show that a solution of a𝑦
= 0) at most once (i.e., y(x) only meets x-axis just once or does not meet).
7. (10 points) Find the general solution of the given differential equation. g(x) is an arbitrary
continuous function. Leave integral if you cannot integrate.
𝑦
5𝑦
6𝑦
𝑔 𝑥
8. (10 points) Find the general solution of the given differential equation.
𝑦
2𝑦
y
2𝑒
1