From Example 5.6.2, the general solution of the given differential equation can be written in the implicit form $y^2\=\sqrt{1\+x^2}\+c$.

Applying the initial condition $y\left(1\right)\=2$ to this solution results in the equation

Now, to obtain $y\left(x\right)$ explicitly requires some care.
Taking the square root leads to

This solution is graphed in Figure 5.6.3(a).

Control-drag the differential equation.
Press the Enter key.

Context Panel:
Add an Initial Condition≻$y\left(1\right)\=2$
(See Figure 5.6.3(b).)

Context Panel: Solve DE≻$y\left(x\right)$ 

Context Panel: Right-hand Side

Context Panel: Plot Builder≻2-D plot
Set $-3\le x\le 3$ 
view: axis[2]≻$\left[0\,2.5\right]$
2-D Options: scaling≻constrained

Tools≻Load Package: Student ODEs

Control-drag the ODE.
Form a set with the ODE and the initial condition.

Apply the ODESteps command.
(This option is not available in the Context Panel.)

$\begin{array}{lll}& & \text{Let's solve}\\ & & \left\{x-2y\left(x\right)\sqrt{x^2+1}\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=0\,y\left(1\right)=2\right\}\\ \text{•}& & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{\ⅆ}{\ⅆx}y\left(x\right)\\ \text{•}& & \text{Separate variables}\\ & & \left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)y\left(x\right)=\frac{x}{2\sqrt{x^2+1}}\\ \text{•}& & \text{Integrate both sides with respect to}x\\ & & \int \left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)y\left(x\right)\ⅆx=\int \frac{x}{2\sqrt{x^2+1}}\ⅆx+\mathrm{C1}\\ \text{•}& & \text{Evaluate integral}\\ & & \frac{{y\left(x\right)}^2}{2}=\frac{\sqrt{x^2+1}}{2}+\mathrm{C1}\\ \text{•}& & \text{Solve for}y\left(x\right)\\ & & \left\{y\left(x\right)=\sqrt{\sqrt{x^2+1}+2\mathrm{C1}}\,y\left(x\right)=-\sqrt{\sqrt{x^2+1}+2\mathrm{C1}}\right\}\\ \text{•}& & \text{Use initial condition}y\left(1\right)=2\\ & & 2=\sqrt{\sqrt{2}+2\mathrm{C1}}\\ \text{•}& & \text{Solve for}\mathrm{\_C1}\\ & & \mathrm{C1}=-\frac{\sqrt{2}}{2}+2\\ \text{•}& & \text{Substitute}\mathrm{\_C1}=-\frac{\sqrt{2}}{2}+2\text{into general solution and simplify}\\ & & y\left(x\right)=\sqrt{\sqrt{x^2+1}-\sqrt{2}+4}\\ \text{•}& & \text{Use initial condition}y\left(1\right)=2\\ & & 2=-\sqrt{\sqrt{2}+2\mathrm{C1}}\\ \text{•}& & \text{Solution does not satisfy initial condition}\\ \text{•}& & \text{Solution to the IVP}\\ & & y\left(x\right)=\sqrt{\sqrt{x^2+1}-\sqrt{2}+4}\end{array}$

Control-drag the differential equation.
Context Panel: Solve DE Interactively

The first pane of the  assistant is shown in Figure 5.6.8. The Context Panel automatically inserts the differential equation into the Assistant.

To add the initial condition, click the Edit button at the bottom of the "Conditions" window.

Figure 5.6.3(d) shows the Edit Conditions pane. Fill in the data under "Add Condition", then click both the Add and the Done buttons.

Press the "Solve Symbolically" button to obtain the "Solve Symbolically" pane shown in Figure 5.6.3(f). Click the Solve button to obtain the analytic solution of the IVP.

Figure 5.6.3(e) shows the Plot Options pane, launched by clicking the Plot Options button in the Solve Symbolically pane. Change the default values for the range of the independent variable.

Press the Plot button in the Solve Symbolically pane to obtain a graph of the solution.