Find y as a function of x?

y''' - 14y″ + 48y′ = 70.e^(x)

Characteristic equation without second member:

r³ - 14r² + 48r = 0

r.[r² - 14r + 48] = 0

r.[r² - (8r + 6r) + 48] = 0

r.[r² - 8r - 6r + 48] = 0

r.[(r² - 8r) - (6r - 48)] = 0

r.[r.(r - 8) - 6.(r - 8)] = 0

r.(r - 8).(r - 6) = 0 → the roots are: 0 ; 6 ; 8

y = C₁.e^(r₁.x) + C₂.e^(r₂.x) + C₃.e^(r₃.x) → it gives us:

y = C₁.e^(0) + C₂.e^(8x) + C₃.e^(6x) → you can simplify, recall: e^(0) = 1

y = C₁ + C₂.e^(8x) + C₃.e^(6x) ← this is the general solution

Equation with second member

y = C₄.e^(x)

y' = C₄.e^(x)

y'' = C₄.e^(x)

y''' = C₄.e^(x)

y''' - 14y″ + 48y′ = 70.e^(x)

C₄.e^(x) - 14.C₄.e^(x) + 48.C₄.e^(x) = 70.e^(x)

35.C₄.e^(x) = 70.e^(x)

35.C₄ = 70

C₄ = 2

Recall: y = C₄.e^(x)

y = 2.e^(x) ← this is a particular solution with second member

...then you sum the general solution with the particular solution, and you obtain:

y = C₁ + C₂.e^(8x) + C₃.e^(6x) + 2.e^(x) ← memorize this result

y' = 8.C₂.e^(8x) + 6.C₃.e^(6x) + 2.e^(x)

y'' = 64.C₂.e^(8x) + 36.C₃.e^(6x) + 2.e^(x)

y'' = 512.C₂.e^(8x) + 216.C₃.e^(6x) + 2.e^(x)

Now, you must calculate C₁, C₂ and C₃ (3 unknew, so you need 3 conditions).

First condition:

y = C₁ + C₂.e^(8x) + C₃.e^(6x) + 2.e^(x) → where: y(0) = 25

C₁ + C₂ + C₃ + 2 = 25

C₁ + C₂ + C₃ = 23

Second condition:

y' = 8.C₂.e^(8x) + 6.C₃.e^(6x) → where: y′(0) = 12

8.C₂ + 6.C₃ + 2 = 12

8.C₂ = 10 - 6.C₃

64.C₂ = 80 - 48.C₃

Third condition:

y'' = 64.C₂.e^(8x) + 36.C₃.e^(6x) + 2.e^(x) → where: y'′(0) = 25

64.C₂ + 36.C₃ + 2 = 25

64.C₂ + 36.C₃ = 23 → recall: 64.C₂ = 80 - 48.C₃

80 - 48.C₃ + 36.C₃ = 23

- 12.C₃ = - 57

C₃ = 57/12

→ C₃ = 19/4

Recall: 64.C₂ = 80 - 48.C₃

64.C₂ = 80 - 48.(19/4)

64.C₂ = - 148

C₂ = - 148/64

→ C₂ = - 37/16

Recall: C₁ + C₂ + C₃ = 23

C₁ = 23 - C₂ - C₃

C₁ = 23 - (- 37/16) - (19/4)

C₁ = (368 + 37 - 76)/16

→ C₁ = 329/16

Recall the memorized result:

y = C₁ + C₂.e^(8x) + C₃.e^(6x) + 2.e^(x) → you know C₁, C₂ and C₃

y = (329/16) - (37/16).e^(8x) + (19/4).e^(6x) + 2.e^(x) ← this is the solution