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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)`
Solution
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.
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