Question

Dy/dx= y^4 and y(2)= -1. Y(-1)=

Answer 1

It looks like you're asked to find the value of y(-1) given its implicit derivative,

`(dy)/(dx) = y^4`

and with initial condition y(2) = -1.

The differential equation is separable:

`(dy)/(y^4) = dx`

Integrate both sides:

`\displaystyle \int (dy)/(y^4) = \int dx`

`-\frac1{3y^3} = x + C`

Solve for y :

`\frac1{3y^3} = -x + C`

`3y^3 = \frac1{-x+C} = -\frac1{x + C}`

`y^3 = -\frac1{3x+C}`

`y = -\frac1{\sqrt[3]{3x+C}}`

Use the initial condition to solve for C :

`y(2) = -1 \implies -1 = -\frac1{\sqrt[3]{3*2+C}} \implies C = -5`

Then the particular solution to the differential equation is

`y(x) = -\frac1{\sqrt[3]{3x-5}}`

and so

`y(-1) = -\frac1{\sqrt[3]{3*(-1)-5}} = \boxed{\frac12}`