Question

Hi there, I am having issues solving the following question from calc-12

Answer 1

We have to find the values of constants A and B.

Given the equality in the question, we can write:

`8xe^(6x)=(d)/(dx)[(Ax+B)e^(6x)]`

We then can calculate the derivative as:

`\begin{gathered} (d)/(dx)[(Ax+B)e^(6x)] \\ A\cdot(d)/(dx)(xe^(6x))+B\cdot(d)/(dx)(e^(6x)) \\ A(x\cdot6e^(6x)+e^(6x))+B(6e^(6x)) \\ (6Ax+A+6B)e^(6x) \end{gathered}`

Then, we can compare the terms:

`\begin{gathered} (8x)e^(6x)=(6Ax+A+6B)e^(6x) \\ \Rightarrow8=6A \\ \Rightarrow0=A+6B \end{gathered}`

We compared the linear and independent terms to find equations to solve for A and B.

Then, we can find A as:

`8=6A\Rightarrow A=(8)/(6)=(4)/(3)`

and B as:

`\begin{gathered} 0=A+6B \\ 6B=-A \\ 6B=-(4)/(3) \\ B=-(4)/(6*3) \\ B=-(4)/(18) \\ B=-(2)/(9) \end{gathered}`

Answer: A = 4/3 and B = -2/9.