Question

Question: In Problems 43 and 44 we saw that every autonomous first order differential equation$\mathbf{dy}\mathbf{/}\mathbf{dx}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{y}\mathbf{\right)}$is separable. Does this fact help in the solution of the initial-value problem$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\sqrt{\mathbf{1}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}{\mathbf{sin}}^{\mathbf{2}}\mathbf{y}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$? Discuss. Sketch, by hand, a plausible solution curve of the problem.

Short answer

Answer:

The sketch can’t be drawn.

Step-by-step solution

Definition

A first-order differential equation of the form$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathit{h}\mathbf{\left(}\mathit{y}\mathbf{\right)}$is said to be separable or to have separable variables.

Separation

Consider the initial value problem$\frac{dy}{dx}=\sqrt{1+y^2}{Sin}^{2}y,y\left(0\right)=\frac{1}{2}.$

Use the separable method to find the general solution of the given initial value problem as follows:

Consider,$\frac{dy}{dx}=\sqrt{1+y^2}{Sin}^{2}y$

$\frac{dy}{\sqrt{1+y^2}{Sin}^{2}y}=dx$since use separable method

Integrating on both sides,

$\int \frac{dy}{\sqrt{1+y^2}{Sin}^{2}y}=\int 1dx$

In this case the integralrole="math" localid="1663844085534" $\int \frac{dy}{\sqrt{1+y^2}{Sin}^{2}y}$is not possible.

As the integralrole="math" localid="1663844080243" $\frac{dy}{dx}⩾0$gives a complex functions.

Note that; for all values of x and y.

And $\frac{dy}{dx}=0$when.$y=0 \& y=\pi$

So,$y=0 \& y=\pi$are equilibrium solutions.

Here the differential equation does not contain solution, so we cannot draw the plot.