- 2069 Chaitra
Solve y''+2y'=eˣ+e⁻ˣ
Short answer
The complementary function is c₁ + c₂e^(-2x). The particular integral is eˣ/3 - e^(-x), giving y = c₁ + c₂e^(-2x) + eˣ/3 - e^(-x).
Solve the differential equation d²y/dx² + 2 dy/dx = eˣ + e⁻ˣ.
The equation is (D² + 2D)y = eˣ + e^(−x).
Complementary function
m² + 2m = 0, so m = 0, −2.
CF = c₁ + c₂e^(−2x)
Particular integral
For eˣ: put D = 1: eˣ/(1 + 2) = eˣ/3.
For e^(−x): put D = −1: e^(−x)/(1 − 2) = −e^(−x).
General solution
y = c₁ + c₂e^(−2x) + eˣ/3 − e^(−x)
✓ Answer checked by Claude Opus 5.5 (high effort)
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