Question

Which of the following is a solution of the differential equation xy' + y = 16x?

y = 16x + 8x^−1
y = 16x − 8x^−1
y = 8x − 8x^−1
y = 8x^−1 − 8x
y = 16x^−1 − 8x

Answer 1

The correct solution of the differential equation xy' + y = 16x is y = 16x^(-1) - 8x.

Step-by-step explanation:

To find the solution of the differential equation xy' + y = 16x, we need to solve it by finding the value of y that satisfies the equation. Let's check each of the given options:

1. y = 16x + 8x-1
   This option is not a solution of the differential equation.
2. y = 16x - 8x-1
   This option is not a solution of the differential equation.
3. y = 8x - 8x-1
   This option is not a solution of the differential equation.
4. y = 8x-1 - 8x
   This option is not a solution of the differential equation.
5. y = 16x-1 - 8x
   This option is a solution of the differential equation. When we substitute it into the equation and simplify, we get (16x-1 - 8x)x' + (16x-1 - 8x) = 16x.

Therefore, the correct solution of the differential equation is y = 16x-1 - 8x.

Answer 2

The student's question involves verifying which function is a solution to the differential equation xy' + y = 16x by substitution and simplification.

Step-by-step explanation:

The student is asking to identify which amongst the given functions is a solution to the differential equation xy' + y = 16x. To determine the correct solution, one must substitute each function into the given differential equation and simplify to see if the equation holds true. The correct solution will satisfy the equation for all values of x for which the function is defined.

Upon substitution, if the differential is satisfied, the function is a solution; otherwise, it is not. For example, when substituting y = 16x + 8x⁻¹, we must compute its derivative y' and see if the original equation holds.