Differential Equation Problem

2. 1) Use erf(x):
𝑑𝑦
𝑑𝑥
= 𝑥2 𝑒−𝑥2
Separable DE:
𝑑𝑦 = 𝑥2 𝑒−𝑥2
𝑑𝑥
Integrate:
∫ 𝑑𝑦 = ∫ 𝑥2 𝑒−𝑥2
𝑑𝑥
𝑦 = ∫ 𝑥2 𝑒−𝑥2
𝑑𝑥
𝑦 = ∫( 𝑥) 𝑥𝑒−𝑥2
𝑑𝑥 [ 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛 𝐵𝑦 𝑃𝑎𝑟𝑡𝑠]
Derivative Integral
+
𝑥
-∫
1
−
1
2
𝑒−𝑥2
𝑦 = −
1
2
𝑥 𝑒−𝑥2
+
1
4
√ 𝜋 erf( 𝑥)
Solution:
𝑦 =
1
4
[√ 𝜋 erf(𝑥) − 2𝑥𝑒−𝑥2
] + 𝐶

3. 2) Use 𝐽( 𝑥):
2
𝜋
∫
sin(𝑡)
𝑡
𝑑𝑡:
𝑥
0
𝑑𝑦
𝑑𝑥
=
sin(𝑥)
𝑥𝑦2
Separable DE:
𝑥𝑦2 𝑑𝑦 = sin( 𝑥) 𝑑𝑥
𝑦2 𝑑𝑦 =
sin( 𝑥)
𝑥
𝑑𝑥
Integrate:
∫ 𝑦2 𝑑𝑦 = ∫
sin(𝑥)
𝑥
𝑑𝑥
𝑦3
3
=
𝜋
2
𝐽(𝑥)
𝑦3 =
3𝜋
2
𝐽(𝑥)
[ 𝑦3]1/3 = [
3𝜋
2
𝐽(𝑥)]
1/3
Solution:
𝑦 = [
3𝜋
2
𝐽(𝑥)]
1/3
3) Example of Various DEs:
a. Both Linear and Separable:
𝑦′ = 𝑥( 𝑦 − 𝑎)
Since
Linear:
𝑦′ = 𝑥𝑦 − 𝑥𝑎
𝑦′ − 𝑥𝑦 = −𝑥𝑎
Separable:
𝑦′
(𝑦 − 𝑎)
= 𝑥
b. Linear but not Separable:
𝑦′ − 𝑥𝑦 = 𝑡