1. Differential Equations: Separation of Variables
Silent
Teacher
Intelligent Practice
Narration
Your Turn
𝑑𝑦 𝑑𝑥 =𝑥𝑦
𝑦 𝑑𝑦 𝑑𝑥 =𝑥
𝑑𝑦 𝑑𝑥 = 1 𝑥𝑦
Practice

2. 𝑑𝑦 𝑑𝑥 = 𝑥 2 𝑦 𝑑𝑦 𝑑𝑥 = 𝑦 2 𝑥 1 𝑦 2 𝑑𝑦 = 𝑥 𝑑𝑥 − 1 𝑦 = 𝑥 2 2 +𝑐
Worked Example
Your Turn
Find a general solution to
Find a general solution to
𝑑𝑦 𝑑𝑥 = 𝑥 2 𝑦
𝑑𝑦 𝑑𝑥 = 𝑦 2 𝑥
1 𝑦 2 𝑑𝑦 = 𝑥 𝑑𝑥
− 1 𝑦 = 𝑥 𝑐
𝑦= 2 𝐴− 𝑥 2
where 𝐴 is a constant

3. 7. 𝑑𝑦 𝑑𝑥 =2𝑥𝑦 1.𝑦 𝑑𝑦 𝑑𝑥 =2𝑥 8. 𝑑𝑦 𝑑𝑥 =2𝑥 𝑦 2 2. 2𝑥 𝑑𝑦 𝑑𝑥 =𝑦
Find the general solution to each differential equation in the form 𝒚=𝒇(𝒙)
7. 𝑑𝑦 𝑑𝑥 =2𝑥𝑦
8. 𝑑𝑦 𝑑𝑥 =2𝑥 𝑦 2
9. 𝑥 2 𝑑𝑦 𝑑𝑥 =2 𝑦
𝑥 𝑑𝑦 𝑑𝑥 = 𝑦 2
11. 𝑦 2 𝑑𝑦 𝑑𝑥 = 𝑥
12. 𝑑𝑦 𝑑𝑥 = 𝑥𝑦
1.𝑦 𝑑𝑦 𝑑𝑥 =2𝑥
2. 2𝑥 𝑑𝑦 𝑑𝑥 =𝑦
3. 2 𝑥 𝑑𝑦 𝑑𝑥 = 1 𝑦
4. 2 𝑥 𝑑𝑦 𝑑𝑥 =𝑦
5. 2 𝑥 𝑑𝑥 𝑑𝑦 =𝑦
6. 1 𝑥𝑦 𝑑𝑦 𝑑𝑥 = 1 2

4. 1. sin 2 𝑦 𝑑𝑦 𝑑𝑥 = cos 𝑥 7. 𝑑𝑦 𝑑𝑥 =2𝑥 𝑒 𝑦 2. sec 2 𝑦 𝑑𝑦 𝑑𝑥 = cosec 𝑥
Find the general solution to each differential equation
1. sin 2 𝑦 𝑑𝑦 𝑑𝑥 = cos 𝑥
2. sec 2 𝑦 𝑑𝑦 𝑑𝑥 = cosec 𝑥
3. sec 𝑦 𝑑𝑦 𝑑𝑥 =2 cosec 2 𝑥
4. cosec 𝑥 𝑑𝑦 𝑑𝑥 =2 sec 𝑦
5. 𝑑𝑦 𝑑𝑥 =2 sec 𝑥 tan 𝑦
6. 𝑑𝑦 𝑑𝑥 =2 sec 𝑦 tan 𝑥
7. 𝑑𝑦 𝑑𝑥 =2𝑥 𝑒 𝑦
8. 𝑑𝑦 𝑑𝑥 =2𝑥 𝑒 𝑥+𝑦
9. 𝑑𝑦 𝑑𝑥 = 2 𝑦 𝑒 𝑥+𝑦
10. 𝑑𝑦 𝑑𝑥 = ln 2𝑥 𝑦
11. 𝑑𝑦 𝑑𝑥 = 𝑥 2ln 𝑦
12. 𝑑𝑦 𝑑𝑥 = 2 ln 𝑦

5. 7. 𝑑𝑦 𝑑𝑥 =2𝑥𝑦 1.𝑦 𝑑𝑦 𝑑𝑥 =2𝑥 8. 𝑑𝑦 𝑑𝑥 =2𝑥 𝑦 2 2. 2𝑥 𝑑𝑦 𝑑𝑥 =𝑦
Find the general solution to each differential equation in the form 𝒚=𝒇(𝒙)
7. 𝑑𝑦 𝑑𝑥 =2𝑥𝑦
8. 𝑑𝑦 𝑑𝑥 =2𝑥 𝑦 2
9. 𝑥 2 𝑑𝑦 𝑑𝑥 =2 𝑦
𝑥 𝑑𝑦 𝑑𝑥 = 𝑦 2
11. 𝑦 2 𝑑𝑦 𝑑𝑥 = 𝑥
12. 𝑑𝑦 𝑑𝑥 = 𝑥𝑦
1.𝑦 𝑑𝑦 𝑑𝑥 =2𝑥
2. 2𝑥 𝑑𝑦 𝑑𝑥 =𝑦
3. 2 𝑥 𝑑𝑦 𝑑𝑥 = 1 𝑦
4. 2 𝑥 𝑑𝑦 𝑑𝑥 =𝑦
5. 2 𝑥 𝑑𝑥 𝑑𝑦 =𝑦
6. 1 𝑥𝑦 𝑑𝑦 𝑑𝑥 = 1 2
𝑦 2 =2 𝑥 2 +𝐴
𝑦=± 2𝑥 2 +𝐴
ln 𝑦 = 𝑥 2 +𝑐
𝑦=𝐴 𝑒 𝑥 2
ln 𝑦 = 1 2 ln 𝑥 + ln 𝐴
𝑦=𝐴 𝑥
− 1 𝑦 = 𝑥 2 +𝑐
𝑦= 1 𝐴− 𝑥 2
𝑦 2 = 𝑥 𝐴
𝑦=± 𝑥 𝐴
𝑦 =− 1 𝑥 +𝑐
𝑦= 1 𝑥 2 − 2𝑐 𝑥 + 𝑐 2
− 1 𝑦 = 𝑥 +𝑐
𝑦= 1 𝐴− 𝑥
ln 𝑦 = 𝑥 𝑐
𝑦=𝐴 𝑒 𝑥 2 `4
𝑦 3 3 = 𝑥 3 +𝑐
𝑦= 2 𝑥 3 +𝐴 1 3
𝑦=±2 ln 𝐴𝑥
ln 𝑦 = 𝑥 𝑐
𝑦=𝐴 𝑒 𝑥 2 `4
2 𝑦 = 𝑥 3 +𝑐
𝑦= 𝑥 3 +𝐴 2 9

6. 1. sin 2 𝑦 𝑑𝑦 𝑑𝑥 = cos 𝑥 7. 𝑑𝑦 𝑑𝑥 =2𝑥 𝑒 𝑦 2. sec 2 𝑦 𝑑𝑦 𝑑𝑥 = cosec 𝑥
Find the general solution to each differential equation
− 1 𝑒 𝑦 = 𝑥 2 +𝑐
𝑦= ln 1 𝐴− 𝑥 2
1. sin 2 𝑦 𝑑𝑦 𝑑𝑥 = cos 𝑥
2. sec 2 𝑦 𝑑𝑦 𝑑𝑥 = cosec 𝑥
3. sec 𝑦 𝑑𝑦 𝑑𝑥 =2 cosec 2 𝑥
4. cosec 𝑥 𝑑𝑦 𝑑𝑥 =2 sec 𝑦
5. 𝑑𝑦 𝑑𝑥 =2 sec 𝑥 tan 𝑦
6. 𝑑𝑦 𝑑𝑥 =2 sec 𝑦 tan 𝑥
7. 𝑑𝑦 𝑑𝑥 =2𝑥 𝑒 𝑦
8. 𝑑𝑦 𝑑𝑥 =2𝑥 𝑒 𝑥+𝑦
9. 𝑑𝑦 𝑑𝑥 = 2 𝑦 𝑒 𝑥+𝑦
10. 𝑑𝑦 𝑑𝑥 = ln 2𝑥 𝑦
11. 𝑑𝑦 𝑑𝑥 = 𝑥 2ln 𝑦
12. 𝑑𝑦 𝑑𝑥 = 2 ln 𝑦
𝑦 2 − 1 2 sin 2𝑦 = sin 𝑥 +𝑐
− 1 𝑒 𝑦 =2𝑥 𝑒 𝑥 −2 𝑒 𝑥 +𝑐
𝑒 𝑦 = 1 2 𝑒 𝑥 −2𝑥 𝑒 𝑥 +𝐴
tan 𝑦 = ln 𝐴 − ln | cosec 𝑥+ cot 𝑥|
tan 𝑦= ln 𝐴 cosec 𝑥+ cot 𝑥
−𝑦 𝑒 −𝑦 − 𝑒 −𝑦 =2 𝑒 𝑥 +𝑐
𝑒 −𝑦 𝑦+1 =2 𝑒 𝑥 +𝑐
ln sec 𝑦 + tan 𝑦 =− 2cot 𝑥 +𝑐
𝑦 2 2 =𝑥 ln 2𝑥+𝑥+𝑐
𝑦=± 2𝑥 ln 2𝑥 +2𝑥+𝐴
sin 𝑦=−2 cos 𝑥 +𝑐
𝑦 ln 𝑦+𝑦 = 𝑥 𝑐
ln sin 𝑦 = ln sec 𝑦 + tan 𝑦 + ln 𝐴
ln sin 𝑦 = ln (𝐴( sec 𝑦+ tan 𝑦 ) )
𝑦 ln 𝑦+𝑦 =2𝑥+𝑐
sin 𝑦 = ln sec 𝑥 + ln 𝐴
sin 𝑦 = ln (𝐴 sec 𝑥 )