Identify the statement(s) which is/are True. | vedclass

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(D) Step $1$: Check option $A$. $A$ function $f(x, y)$ is homogeneous of degree $n$ if $f(\lambda x, \lambda y) = \lambda^n f(x, y)$. For $f(x, y) = e

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All of the above.

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(D) Step $1$: Check option $A$. $A$ function $f(x, y)$ is homogeneous of degree $n$ if $f(\lambda x, \lambda y) = \lambda^n f(x, y)$. For $f(x, y) = e^{y/x} + 	an(y/x)$,$f(\lambda x, \lambda y) = e^{(\lambda y)/(\lambda x)} + 	an((\lambda y)/(\lambda x)) = e^{y/x} + 	an(y/x) = \lambda^0 f(x, y)$. Thus,it is homogeneous of degree $0$. Option $A$ is true.Step $2$: Check option $B$. $A$ differential equation $M(x, y)dx + N(x, y)dy = 0$ is homogeneous if both $M(x, y)$ and $N(x, y)$ are homogeneous functions of the same degree. Here,$M(x, y) = x \ln(y/x)$ is homogeneous of degree $1$ (since $M(\lambda x, \lambda y) = \lambda x \ln(y/x) = \lambda^1 M(x, y)$). $N(x, y) = \frac{y^2}{x} \sin^{-1}(y/x)$ is homogeneous of degree $1$ (since $N(\lambda x, \lambda y) = \frac{(\lambda y)^2}{\lambda x} \sin^{-1}(y/x) = \lambda \frac{y^2}{x} \sin^{-1}(y/x) = \lambda^1 N(x, y)$). Since both are degree $1$,the equation is homogeneous. Option $B$ is true.Step $3$: Check option $C$. For $f(x, y) = x^2 + \sin x \cos y$,$f(\lambda x, \lambda y) = (\lambda x)^2 + \sin(\lambda x) \cos(\lambda y) = \lambda^2 x^2 + \sin(\lambda x) \cos(\lambda y)$. This cannot be written in the form $\lambda^n f(x, y)$ for any $n$. Thus,it is not homogeneous. Option $C$ is true.Step $4$: Since $A, B,$ and $C$ are true,the correct answer is $D$.

Identify the statement$(s)$ which is/are True.

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