Problema1.docx

Resolver:

𝑦 ǀǀǀ − 4𝑦 ǀǀ + 5𝑦 ǀ = 4𝑥

𝑦 = 𝑦𝑔 + 𝑦𝑝 Solución:

𝑚3 − 4𝑚2 + 5𝑚 = 0 𝑚(𝑚2 − 4𝑚 + 5) = 0 𝑚(𝑚 − 5)(𝑚 + 1) = 0 𝑚1 = 0 , 𝑚2 = 5, 𝑚3 = −1 𝑦𝑔 = 𝑐1 + 𝑐2 𝑒5𝑥 +𝑐3 𝑒−𝑥

Calculando 𝑦𝑝 : 𝑢1 + 𝑢2 𝑒 5𝑥 + 𝑢3 𝑒 −𝑥 1 𝑒 5𝑥 𝑤 = |0 5𝑒 5𝑥 0 25𝑒 5𝑥

0 𝑤1 = | 0 4𝑥 1 𝑤2 = |0 0

𝑒 −𝑥 −𝑒 −𝑥 | = −20𝑒 4𝑥 𝑒 −𝑥

𝑒 5𝑥 5𝑒 5𝑥 25𝑒 5𝑥 0 0 4𝑥

𝑒 −𝑥 −𝑒 −𝑥 | − 24𝑥𝑒 4𝑥 𝑒 −𝑥 𝑒 −𝑥 −𝑒 −𝑥 | = 4𝑥𝑒 −𝑥 𝑒 −𝑥

1 𝑒 5𝑥 𝑤3 = |0 5𝑒 5𝑥 0 25𝑒 5𝑥

0 0 | = 20𝑥𝑒 5𝑥 4𝑥

𝑢1ǀ

−24𝑥𝑒 4𝑥 6𝑥 = = −20𝑒 4𝑥 5

𝑢2ǀ

4𝑥𝑒 −𝑥 −4𝑥𝑒 −5𝑥 = = −20𝑒 4𝑥 20

𝑢3ǀ

20𝑥𝑒 5𝑥 −20𝑥𝑒 𝑥 = = −20𝑒 4𝑥 20

3𝑥 2 𝑢1 = 5 6𝑒 −5𝑥 𝑢2 = 125 𝑢3 = 𝑒 𝑥 (−𝑥 + 1) Solución general: 𝑦 = 𝑦𝑔 + 𝑦𝑝

5𝑥

𝑦 = 𝑐1 + 𝑐2 𝑒 +𝑐3 𝑒 5𝑥

−𝑥

3𝑥 2 6 + + + (−𝑥 + 1) 5 125

𝑦 = 𝑐1 + 𝑐2 𝑒 +𝑐3 𝑒

−𝑥

3𝑥 2 131 + + −𝑥 5 125