Question

If 𝒚_𝒑𝟏 = 𝒙 is a particular solution of 𝒚 ′′ + 𝒚 = 𝒙 and 𝒚_𝒑𝟐 = 𝒙 ^𝟐 − 𝟐 is a particular solution of 𝒚 ′′ + 𝒚 = 𝒙^𝟐, then a particular solution of 𝒚 ′′ + 𝒚 = 𝒙 ^𝟐 + 𝒙 is

Short answer

The particular solution of 𝑦 ′′ + 𝑦 = 𝑥 ² + 𝑥 is 𝑦_𝑝 = 𝑥 ² + 𝑥 − 2

Step-by-step solution

Define the particular solution.

The particular solution of the equation is given by:

𝑦 ′′ + 𝑦 = 𝑥²+ 𝑥

⏟ ⏟

𝑔2(𝑥) 𝑔1(𝑥)

Solve the equation

If 𝑦_𝑝1 = 𝑥 is the particular solution of

𝑦 ′′ + 𝑦 = 𝑥

and if 𝑦_𝑝2 = 𝑥 ²− 2 is the particular solution of

𝑦 ′′ + 𝑦 = 𝑥 ²

then, it follows from (13) of Theorem 4.1.7 that the particular solution of

𝑦 ′′ + 𝑦 = 𝑥²+ 𝑥

⏟ ⏟

𝑔2(𝑥) 𝑔1(𝑥)

is the superposition of 𝑦_𝑝2 and 𝑦_𝑝1 , i.e.,

𝑦 = 𝑦_𝑝1 + 𝑦_𝑝2

= 𝑥 + 𝑥² − 2

= 𝑥² + 𝑥 − 2

The particular solution of the equation is: 𝑦_𝑝 = 𝑥² + 𝑥 − 2