Question

Solve the equation e2x dy/dx 1 given that y = 5 when x = 0 ​

Answer 1

The equation is

`{ \underline{ \sf{y = - 2 {e}^( - 2x) + 7 }}}`

Explanation:

`{ \sf{ {e}^(2x) (dy)/(dx) = 1}} \\ \\ { \sf{dy = {e}^( - 2x) dx}}`

integrate:

`{ \sf{ \int dy = \int { - e}^(2x) dx}} \\ { \sf{y = - 2 {e}^( - 2x) + c}}`

c is a constant.

when y is 5, x is 0:

`{ \sf{y = { - 2e}^( - 2x) + c}} \\ { \sf{5 = - 2 {e}^(( - 2 * 0)) + c }} \\ { \sf{5 = - 2 {e}^(0) + c }} \\ { \sf{5 = ( - 2 * 1) + c}} \\ { \sf{5 = - 2 + c}} \\ { \sf{c = 7}}`

therefore, equation:

`y = - 2 {e}^( - 2x) + 7`