Domain & Interval of Solution for y=x+4\sqrt{x+2}

[find_the_fun]

The question asks
Verify that the indicated function $$y=\phi(x)$$ is an explicit solution of the given first order differential equation. Consider $$\phi$$ simply as a function, give its domain. Then by considering $$\phi $$as a solution of the differential equation, give at least one interval $$I$$ of definition.

$$(y-x)y'=y-x+8$$; [math]y=x+4\sqrt{x+2}[/math]The first step in solving a problem is understanding the question.

Verify that the indicated function $$y=\phi(x)$$ is an explicit solution of the given first order differential equation.

Understood.

Consider $$\phi$$ simply as a function, give its domain.

Is it asking where the function [math]x+4\sqrt{x+2}[/math] is defined?

Then by considering $$\phi $$as a solution of the differential equation, give at least one interval $$I$$ of definition.

What is an interval of definition?So I took the derivative of y and was verifying the solution and got to the equation

[math]4\sqrt{x+2}+\frac{16\sqrt{x+2}}{2\sqrt{x+2}}=4\sqrt{x+2}[/math] and got a little worried. This equality is only true when x=-2.

The answer key gives: domain of fuctnion is $$[-2, \infty)$$; largest interval of definition for solution is $$(-2, \infty)$$

 

[Prove It]

The question asks
Verify that the indicated function $$y=\phi(x)$$ is an explicit solution of the given first order differential equation. Consider $$\phi$$ simply as a function, give its domain. Then by considering $$\phi $$as a solution of the differential equation, give at least one interval $$I$$ of definition.

$$(y-x)y'=y-x+8$$; [math]y=x+4\sqrt{x+2}[/math]The first step in solving a problem is understanding the question.Understood.Is it asking where the function [math]x+4\sqrt{x+2}[math] is defined?
What is an interval of definition?So I took the derivative of y and was verifying the solution and got to the equation

[math]4\sqrt{x+2}+\frac{16\sqrt{x+2}}{2\sqrt{x+2}}=4\sqrt{x+2}[/math] and got a little worried. This equality is only true when x=-2.

The answer key gives: domain of fuctnion is $$[-2, \infty)$$; largest interval of definition for solution is $$(-2, \infty)$$

Wait, what is your $\displaystyle \begin{align*} \phi \end{align*}$ function that you are trying to test?

 

[find_the_fun]

Wait, what is your $\displaystyle \begin{align*} \phi \end{align*}$ function that you are trying to test?

I don't really understand what $\phi$ is?

 

[I like Serena]

Hi find_the_fun,

The question asks
Verify that the indicated function $$y=\phi(x)$$ is an explicit solution of the given first order differential equation. Consider $$\phi$$ simply as a function, give its domain. Then by considering $$\phi $$as a solution of the differential equation, give at least one interval $$I$$ of definition.

$$(y-x)y'=y-x+8$$; [math]y=x+4\sqrt{x+2}[/math]

Consider ϕ simply as a function, give its domain.

Is it asking where the function [math]x+4\sqrt{x+2}[/math] is defined?

Yes.

What is an interval of definition?

The interval of definition is the interval for $x$ where the differential equation is properly defined.
In particular it means that $y'$ must be properly defined.

So I took the derivative of y and was verifying the solution and got to the equation

[math]4\sqrt{x+2}+\frac{16\sqrt{x+2}}{2\sqrt{x+2}}=4\sqrt{x+2}[/math] and got a little worried. This equality is only true when x=-2.

You should have an extra $+8$ on the right hand side.