Differential-Equations Integrating Differential Equations, guaranteed and verified 100% Pass

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Integrating Differential Equations


A differential equation is an equation relating a function and one or more of its
derivatives. The solutions to a differential equation are all functions that satisfy
the equation.
Differential Equation General Solution
𝑑𝑦
= 2𝑥 𝑦 = 𝑥2 + 𝑐
𝑑𝑥

𝑦 ′′ + 9𝑦 = 0 𝑦 = 𝐴 cos 3𝑥 + 𝐵 sin 3𝑥

In the second case notice that:

𝑦 = 𝐴 cos 3𝑥 + 𝐵 sin 3𝑥
𝑦 ′ = −3𝐴 sin 3𝑥 + 3𝐵 cos 3𝑥
𝑦 ′′ = −9𝐴 cos 3𝑥 − 9𝐵 sin 3𝑥

Thus we have:
𝑦 ′′ + 9𝑦 = (−9𝐴 cos 3𝑥 − 9𝐵 sin 3𝑥) + 9(𝐴 cos 3𝑥 + 𝐵 sin 3𝑥) = 0.

Hence 𝑦 = 𝐴 cos 3𝑥 + 𝐵 sin 3𝑥 is a solution to the differential equation

𝑦 ′′ + 9𝑦 = 0.


𝑦 ′′ + 9𝑦 = 0 is called a 2nd order differential equation because the
second deriative of 𝑦 appears in the equation.

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Initial Value Problem: The solution to a differential equation given an initial
condition or conditions.


𝑑𝑦
Ex. Solve = 4𝑥 3 ; with the initial condition that 𝑦(1) = 3.
𝑑𝑥



General Solution: 𝑦 = 𝑥4 + 𝑐
Initial Condition: 𝑦(1) = 14 + 𝑐 = 3

⇒𝑐=2
𝑦 = 𝑥 4 + 2 is called the Particular Solution.
𝑦 = 𝑥 4 + 𝑐 is called the General Solution.


𝑑𝑦 𝑑2𝑦
If we have = 𝑔(𝑥) or 𝑑𝑥 2 = 𝑔(𝑥) we can try integrating both sides.
𝑑𝑥



𝑑𝑦
Ex. Solve = 𝑥 2 √𝑥 3 + 1 ; 𝑦(2) = 3.
𝑑𝑥

General Solution:
1
𝑦 = ∫ 𝑥 2 (𝑥 3 + 1)2 𝑑𝑥
Let 𝑢 = 𝑥 3 + 1

𝑑𝑢 = 3𝑥 2 𝑑𝑥
1
𝑑𝑢 = 𝑥 2 𝑑𝑥
3
1 1
1 1 2 3
∫ 𝑥 2 (𝑥 3 + 1) 𝑑𝑥 =
2 ∫ 𝑢2 𝑑𝑢 = ( 𝑢2 ) + 𝑐
3 3 3
3
2 3
𝑦= (𝑥 + 1) + 𝑐
2
9