Question

In the following verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:-

y = x sin x              `xy'=y+xsqrt(x^2-y^2)`

Answer 1

We have,

`xy'=y+xsqrt(x^2-y^2)  ................(1)`

Now,

y = x sin x

`rArry'=sinx + xcosx`

Putting the above value in (1), we get

LHS = x (sin x + x cos x)

= x sin x + x² cos x

= x sin x + x(x cos x)

`=xsinx+x(xsqrt(1-sin^2x))`

`=xsinx+x(x^2-x^2sin^2x)`

`=y+x(sqrt(x^2-y^2)="RHS"`

Thus, y= x sin x is the solution of the given differential equation.