Question

In Exercises \(11-20,\) solve the initial value problem explicitly. $$\frac{d A}{d x}=10 x^{9}+5 x^{4}-2 x+4\( and \)A=6\( when \)x=1$$

Short answer

The solution to the initial value problem is \( A(x) = x^{10} + x^{5} - x^{2} + 4x + 1 \).

Step-by-step solution

Integrate the differential equation

The differential equation given is : \(\frac{d A}{d x}=10 x^{9}+5 x^{4}-2 x+4\). Its integral is:\[A(x) = \int (10 x^{9}+5x^{4}-2x+4) dx = x^{10} + x^{5} - x^{2} + 4x + C\]where C is the constant of integration that we will determine using the initial value.

Apply the initial condition to find the constant C

The initial condition is A=6 when x=1. Plugging these values into the equation we will find C:\[6 = 1^{10} + 1^{5} - 1^{2} + 4*1 + C = 5 + C \Rightarrow C = 6 - 5 = 1\].

Give the complete solution

Now we substitute the constant C back into the integrated equation: \[ A(x) = x^{10} + x^{5} - x^{2} + 4x + 1\].