Question

Find a singular solution of Problem$\mathbf{21}$& problem$\mathbf{22}$

Short answer

The singular solution of problem 21 & 22 are $y=\pm 1$and $y=0$respectively.

Step-by-step solution

Definition

Singular solution is solution of a differential equation that cannot be obtained from the general solution gotten by the usual method of solving the differential equation.

Check for singular solution

Problem 21 and 22 are 21.$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x}\sqrt{1-{\mathrm{y}}^2}$&22.$(e^x+e^{-x})\frac{dy}{dx}=y^2$.

From problem 21, by separating variables, we have

$\frac{dy}{\sqrt{1-y^2}}=xdx$ ----(1)

Note that, when $y=\pm 1$, the left - hand side of equation (1) is undefined.

Thus, $y=1$and $y=-1$are the corresponding singular solutions.

Check for singular solution

By separating variables in problem 22, we have

$\frac{\mathrm{dy}}{{\mathrm{y}}^2}=\frac{\mathrm{dx}}{{\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{-\mathrm{x}}}\mathrm{..........}\left(2\right)$

Note that, when $y=0$, the left-hand side of equation (2)is undefined.

Thus,$y=0$ is a singular solution.