The particular solution of the differential equation that satisfies the initial condition(s) is [tex]\rm f(s) = 7s^2-3s^2-189[/tex].
Given that,
Differential equation; [tex]\rm f '(s) = 14s - 12s^3[/tex]
We have to determine
The particular solution of the differential equation that satisfies the initial condition(s) f,(3) = 9.
According to the question,
To determine the solution of the differential equation first integrate the equation and then solve the equation.
Differential equation; [tex]\rm f '(s) = 14s - 12s^3[/tex]
Integrate the equation on both sides,
[tex]\rm \int f '(s) ds= \int (14s -12s^3)ds\\\\ \int f '(s) .ds= \int 14s .ds -\int 12s^3.ds\\\\f(s) = 14 \dfrac{s^2}{2}-12 \dfrac{s^4}{4}\\\\f(s) = 7s^2-3s^4+c[/tex]
Then,
[tex]\rm f(3) = 7(3)^2-3(3)^4+c\\\\9 = 7\times 9 - 3\times 81+c\\\\c = -9+63-243\\\\c=-189[/tex]
Therefore,
The particular solution of the differential equation that satisfies the initial condition(s) is,
[tex]\rm f(s) = 7s^2-3s^2-189[/tex]
Hence, The particular solution of the differential equation that satisfies the initial condition(s) is [tex]\rm f(s) = 7s^2-3s^2-189[/tex].
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