Problem Set 1 First order DE (Do at least 5 problems of each type)
1) ๐ฆโฒ= ๐ฆ โ ๐ฅ, ๐ฆ(0)= 1/3
2) ๐ฆโฒ= ๐ฆ4/5, ๐ฆ(0)= 1
3) ๐ฆโฒ=๐ฅ๐ฆ
๐ฅ2+4
4) 3๐ฅ๐ฆโฒโ 1 = ln ๐ฅ + ๐ฆ, ๐ฆ(1)= 2
5) ๐ฆโฒ=1โ2๐ฆโ4๐ฅ
1+๐ฆ+2๐ฅ by using ๐ฆ + 2๐ฅ = ๐ฃ
6) ๐ฅ3๐ฆโฒ + 3๐ฅ2๐ฆ = 1/๐ฅ
7) ๐ฆโฒ= ๐ฆ + ๐ฆ2, ๐ฆ(0)= 0
8) ๐ฆโฒ = 2(๐ฆ โ 1)tanh 2๐ฅ , ๐ฆ(0)= 4
9) Show that ๐ฆ = ๐๐๐ฅ+ ๐ฅ + 1 is the solution to ๐ฆโฒ= ๐ฆ โ ๐ฅ when ๐ฆ(0)= 3
10) ๐ฆโฒ๐ฅ ๐๐ ๐ฅ = ๐ฆ, ๐ฆ(3)=ln 81
11) 2๐ฆ๐ฆโฒ+ ๐ฆ2sin ๐ฅ = sin ๐ฅ, ๐ฆ(0)=โ2
12) ๐ฆโฒ+ 2๐ฆ = ๐ก2โ ๐ก + 1, ๐ฆ(1)= 1
2
13) ๐ฆโฒ=๐ฅ๐ฆ3
โ1+๐ฅ2
14) ๐๐
๐๐ = โ๐ tan ๐
15) ๐๐ฆ
๐๐ฅ =๐ฅ(๐๐ฅ2+2)
6๐ฆ2; ๐ฆ(0)= 1
16) ๐๐ฆ
๐๐ฅ = 2๐ฅ(๐ฆ โ 5)
17) ๐๐ฆ
๐๐ฅ = (1 + ๐โ๐ฅ)(๐ฆ2โ 1)
18) sin ๐ฅ ๐๐ฆ
๐๐ฅ + ๐ฆ cos ๐ฅ = sin ๐ฅ
19) ๐ฅ๐ฆโฒ+ ๐ฆ = ๐ฅ2+ 1
20) ๐๐ฆ
๐๐ฅ + 2๐ฆ = cos ๐ฅ
