A.![y(x) =\sqrt{ 2(\frac{x^5}{5} -\frac{6^5}{5})}](/tpl/images/0524/7025/6834f.png)
B.Therefore the solution is defined on the interval:
6≤ x ≤ +∞
Step-by-step explanation:
Given differential equation is
![\frac{dy}{dx} = x^4y^{-1}](/tpl/images/0524/7025/a8056.png)
![\Rightarrow \frac{dy}{dx} =\frac{ x^4}{y}](/tpl/images/0524/7025/d9791.png)
![\Rightarrow y dy = x^4dx](/tpl/images/0524/7025/265d3.png)
Integrating both sides:
![\Rightarrow\int y dy = \int x^4dx](/tpl/images/0524/7025/741b6.png)
........(1)
Given y(0)= 6
it means y=0 when x=6
Putting x=6 and y=0 in the equation (1)
![\therefore \frac{0^2}{2} =\frac{6^5}{5} +C](/tpl/images/0524/7025/ecbe7.png)
![\Rightarrow C=-\frac{6^5}{5}](/tpl/images/0524/7025/379f6.png)
Equation (1) become:
![\Rightarrow \frac{y^2}{2} =(\frac{x^5}{5} -\frac{6^5}{5})](/tpl/images/0524/7025/b2b69.png)
![\Rightarrow y} =\sqrt{ 2(\frac{x^5}{5} -\frac{6^5}{5})}](/tpl/images/0524/7025/cdacf.png)
Therefore ![y(x) =\sqrt{ 2(\frac{x^5}{5} -\frac{6^5}{5})}](/tpl/images/0524/7025/6834f.png)
(B)
The quantity under root must greater than or equal to zero.
Therefore,
![2(\frac{x^5}{5} -\frac{6^5}{5})} \geq0](/tpl/images/0524/7025/2d456.png)
![\Rightarrow \frac{x^5}{5} -\frac{6^5}{5}} \geq0](/tpl/images/0524/7025/7830a.png)
![\Rightarrow \frac{x^5}{5} \geq\frac{6^5}{5}}](/tpl/images/0524/7025/bfcab.png)
![\Rightarrow x^5\geq {6^5](/tpl/images/0524/7025/af864.png)
![\Rightarrow \sqrt[5]{x^5} \geq \sqrt[5]{6^5}](/tpl/images/0524/7025/306f8.png)
![\Rightarrow x\geq {6](/tpl/images/0524/7025/69923.png)
When x≥0 then the value of
is real other wise the value of
is imaginary.
Therefore the solution is defined on the interval:
6≤ x ≤ +∞