Question

Consider the first order separable equation y′=20x^4y(1+3x^5)^1/3. An explicit general solution can be written in the form y=Cf(x) for some function f(x) with C an arbitrary constant. Here f(x)=_______

Next find the explicit solution of the initial value problem y(0)=3
y=_________ .

Answer 1

`f(x)=Ce^{(1+3x^5)^{(4)/(3)}}`

The explicit solution of the initial value problem=y=
`1.1e^{(1+3x^5)^{(4)/(3)}}`

Explanation:

We are given that first order separable equation

`(dy)/(dx)=20x^4y(1+3x^5)^{(1)/(3)}`

`\int (dy)/(y)=20\int x^4(1+3x^5)^{(1)/(3)}dx`

Suppose
`1+3x^5=t`

Differentiate w.r.t x

`15x^4dx=dt`

`x^4dx=(1)/(15)dt`

Substitute the values

`ln y=(20)/(15)\int t^{(1)/(3)}dt`

`ln y=(4)/(3)* (3)/(4) t^{(4)/(3)}`+C

By using the formula
`\int x^n dx=(x^(n+1))/(n+1)`+C

`\int (dx)/(x)=ln x+C`

`ln y=(1+3x^5)^{(4)/(3)}`+C

`y=e^{(1+3x^5)^{(4)/(3)}+C}`

By using identity :
`ln x=y\implies x=e^y`

`y=e^{(1+3x^5)^{(4)/(3)}}\cdot e^C=Ce^{(1+3x^5)^{(4)/(3)}}`

By using identity :
`x^a\cdot x^y=x^(a+y)`

Where
`e^C=C`

`y=Ce^{(1+3x^5)^{(4)/(3)}}`

By compare with y=Cf(x)

We get
`f(x)=Ce^{(1+3x^5)^{(4)/(3)}}`

We are given that y(0)=3

Substitute the value

`3=Ce`

`C=(3)/(e)=1.1`

Substitute the value

`y=1.1e^{(1+3x^5)^{(4)/(3)}}`

The explicit solution of the initial value problem=y=
`1.1e^{(1+3x^5)^{(4)/(3)}}`