Question

Suppose that y(x)denotes a solution of the first-order IVP$\mathit{y}\mathbf{\text{'}}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{,}\mathit{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}$ and thaty(x)possesses at least a second derivative at x = 1. In some neighbourhood of x = 1use the DE to determine whether y(x)is increasing or decreasing and whether the graph y(x)is concave up or concave down.

Short answer

The function y(x) is an increasing function and its graph is concave down.

Step-by-step solution

Define a derivative of the function

The sensitivity of the function is derived by the derivative of a function of a real variable for modifying the argument.

Calculus uses derivatives as a fundamental tool. When a derivative of a single-variable function exists at a given input value, it is the slope of the tangent line to the function's graph at that point.

Determine whether it is increasing or not

The initial condition$y\left(1\right)=-1$is substituted to the right-hand side of the equation as given by,

. $\begin{array}{c}y\text{'}\left(1\right)=(1{)}^2+(-1{)}^2\\ =1+1\\ =2>0\end{array}$

As $y\text{'}\left(1\right)>0$in some neighbourhood of $x=1$, then y(x) is increasing.

Hence, y(x)is increasing.

Determine the nature of the graph

Differentiate the given equation again to get,

$y\text{'}\text{'}=2x+2yy\text{'}$

The initial condition $y\left(1\right)=-1$is substituted to the right-hand side of the equation with $y\text{'}=2$is given by,

$\begin{array}{c}y\text{'}\text{'}\left(1\right)=2\left(1\right)+2(-1)\left(2\right)\\ =2-4\\ =-2<0\end{array}$

As $y\text{'}\text{'}\left(1\right)>0$in some neighbourhood of $x=1$, then the graph of y(x) is concave down.

Hence, the graph is concave down.