To solve the given initial value problem, we need to follow these steps:
1. Identify the Differential Equation:
The differential equation given is:
[tex]\[
g'(x) = 9x \left(x^8 - \frac{1}{9}\right)
\][/tex]
2. Integrate the Differential Equation:
We need to integrate [tex]\( g'(x) \)[/tex] with respect to [tex]\( x \)[/tex] to find [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = \int 9x \left(x^8 - \frac{1}{9}\right) \, dx
\][/tex]
Breaking the integral into two parts, we get:
[tex]\[
g(x) = 9 \int x \cdot x^8 \, dx - 9 \int x \cdot \frac{1}{9} \, dx
\][/tex]
Simplifying it further:
[tex]\[
g(x) = 9 \int x^9 \, dx - \int x \, dx
\][/tex]
Computing these integrals:
[tex]\[
9 \int x^9 \, dx = 9 \left(\frac{x^{10}}{10}\right) = 0.9 x^{10}
\][/tex]
and
[tex]\[
\int x \, dx = \frac{x^2}{2}
\][/tex]
Therefore:
[tex]\[
g(x) = 0.9 x^{10} - 0.5 x^2 + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
3. Apply the Initial Condition [tex]\( g(1) = 1 \)[/tex]:
To find [tex]\( C \)[/tex], we use the initial condition:
[tex]\[
g(1) = 0.9 (1)^{10} - 0.5 (1)^2 + C = 1
\][/tex]
Simplifying, we get:
[tex]\[
0.9 - 0.5 + C = 1
\][/tex]
[tex]\[
0.4 + C = 1
\][/tex]
[tex]\[
C = 1 - 0.4 = 0.6
\][/tex]
4. Write the Final Solution:
Substituting [tex]\( C = 0.6 \)[/tex] back into the expression for [tex]\( g(x) \)[/tex], we get:
[tex]\[
g(x) = 0.9 x^{10} - 0.5 x^2 + 0.6
\][/tex]
So, the solution to the initial value problem is:
[tex]\[
g(x) = 0.9 x^{10} - 0.5 x^2 + 0.6
\][/tex]
