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Find the particular solution of the differential equation that satisfies the initial conditions. f apostrophe apostrophe open parentheses x close parentheses equals 8 comma space space space space f apostrophe (1 )equals 6 comma space space space space f (0 )equals 5

Respuesta :

Answer:

Therefore the required solution is

[tex]f(x)=4x^2-2x+5[/tex]

Step-by-step explanation:

Rule of integration:

  1. [tex]\int x^ndx=\frac{x^{n+1}}{n+1}+C[/tex]
  2. [tex]\int m\ dx = m\int dx= mx+C[/tex] [ m is a constant]
  3. [tex]\int mx^ndx=m\int x^n dx=m.\frac{x^{n+1}}{n+1}[/tex]

Given that,

f''(x)  = 8 and initial conditions are f'(1)=6 and f(0)=5

∴f''(x)  = 8

Integrating both sides

[tex]\int f''(x) dx=\int 8 dx[/tex]

[tex]\Rightarrow f'(x)= 8x+C_1[/tex]             [ [tex]C_1[/tex] is constant of integration]

Initial condition f'(1) =6

[tex]\therefore 6= 8.1+C_1[/tex]

[tex]\Rightarrow C_1=6-8[/tex]

[tex]\Rightarrow C_1=-2[/tex]

[tex]\therefore f'(x)=8x-2[/tex]

Again integrating both sides

[tex]\int f'(x) dx=\int8x dx- \int2dx[/tex]

[tex]\Rightarrow f(x)=8\frac{x^2}{2}-2x+C_2[/tex]     [ [tex]C_2[/tex] is constant of integration]

[tex]\Rightarrow f(x)=4x^2-2x+C_2[/tex]

Initial condition f(0)=5

[tex]5=4.0^2-2.0+C_2[/tex]

[tex]\Rightarrow C_2=5[/tex]

[tex]\therefore f(x)=4x^2-2x+5[/tex]

Therefore the required solution is

[tex]f(x)=4x^2-2x+5[/tex]

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