The differential equation d y d x =F(x, y) will not be a homogene… - Vidyadip

MCQ

The differential equation $\frac{d y}{d x}=F(x, y)$ will not be a homogeneous differential equation, if $F(x, y)$ is:

A
$\cos x-\sin \left(\frac{y}{x}\right)$
B
$\frac{y}{x}$
C
$\frac{x^2+y^2}{x y}$
D
$\cos ^2\left(\frac{x}{y}\right)$

✓

Answer

$
\begin{array}{l}\text {Let } F(x, y)=\cos x-\sin \frac{y}{x} \\
\Rightarrow F(\lambda x, \lambda y)=\cos \lambda x-\sin \frac{\lambda y}{\lambda x}=\cos \lambda x-\sin \frac{y}{x} \\
\quad \neq \lambda\left(\cos x-\sin \frac{y}{x}\right)
\end{array}
$$\therefore \quad \cos x-\sin \frac{y}{x}$ is not homogeneous.

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