Question

Show that an implicit solution of $\mathbf{2}{\mathbf{xsin}}^2\mathbf{ydx}\mathbf{-}({\mathbf{x}}^2\mathbf{+}\mathbf{10})\mathbf{cosydy}\mathbf{=}\mathbf{0}$

is given by$\mathbf{ln}\mathbf{(}{\mathbf{x}}^2\mathbf{+}\mathbf{10}\mathbf{)}\mathbf{+}\mathbf{cscy}\mathbf{=}\mathbf{c}$. Find the constant solutions, if any, that were lost in the solution of the differential equation.

Short answer

The constant solution is $y=\text{n}\pi ,\text{n}\in \text{Z}$.

Step-by-step solution

Definition

An implicit solution is any solution that isn't in explicit form.

Find implicit solution

By separating variables in$2x{\sin }^{2}ydx-(x^2+10)\mathrm{cosy}dy=0$ .

We have$\frac{\text{cosy}}{{\sin }^{2}y}dy=\frac{2x}{x^2+10}dx\cdots \left(1\right)$

On integration, we obtain,

$-\frac{1}{\sin y}+c={\ln }^{|x^2+10|}$

(Or)${\ln }^{|x^2+10|}+\mathrm{cosec}y=c$

Thus, ${\ln }^{|x^2+10|}+\mathrm{cosecy}=c$; is an implicit solution of given differential equation.

Find constant solution

In equation (1), whenever ${\sin }^{2}y=0$; left-hand side is undefined

$\begin{array}{c}{\sin }^2\mathrm{y}=0\\ \Rightarrow \mathrm{siny}=0=\sin \text{n}\mathrm{\pi }\end{array}$

Thus ,$y=\text{n}\pi ,\text{n}\in \text{Z}$ gives singular solutions.