[tex]\begin{gathered}\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x \: = \: - \: \left[y^{2}-x^{2} \log \left|\frac{y}{x}\right|\right] d y \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x \: = \: \left[x^{2} \log \left|\frac{y}{x}\right| \: - \: {y}^{2} \right] d y \\ \\ \end{gathered} [/tex]
[tex]\begin{gathered}\rm\implies \:\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x+\left[y^{2}-x^{2} \log \left|\frac{y}{x}\right|\right] d y=0 \: is \: homogenous\\ \\ \end{gathered}[/tex]
Alternative Method :-
Given differential equation is
[tex]\begin{gathered}\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x+\left[y^{2}-x^{2} \log \left|\frac{y}{x}\right|\right] d y=0 \\ \\ \end{gathered}[/tex]
can be rewritten as
[tex]\begin{gathered}\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x \: = \: - \: \left[y^{2}-x^{2} \log \left|\frac{y}{x}\right|\right] d y \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x \: = \: \left[x^{2} \log \left|\frac{y}{x}\right| \: - \: {y}^{2} \right] d y \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\sf \: \displaystyle \sf \dfrac{xy}{ {x}^{2} } \log \left|\frac{y}{x}\right| d x \: = \: \left[ \frac{ {x}^{2} }{ {x}^{2} } \log \left|\frac{y}{x}\right| \: - \: \frac{ {y}^{2} }{ {x}^{2} } \right] d y \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\sf \: \displaystyle \sf \dfrac{y}{ x} \log \left|\frac{y}{x}\right| d x \: = \: \left[ \log \left|\frac{y}{x}\right| \: - \: \frac{ {y}^{2} }{ {x}^{2} } \right] d y \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\rm\implies \:\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x+\left[y^{2}-x^{2} \log \left|\frac{y}{x}\right|\right] d y=0 \: is \: homogenous\\ \\ \end{gathered}[/tex]
Answers & Comments
Verified answer
can be rewritten as
[tex]\begin{gathered}\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x \: = \: - \: \left[y^{2}-x^{2} \log \left|\frac{y}{x}\right|\right] d y \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x \: = \: \left[x^{2} \log \left|\frac{y}{x}\right| \: - \: {y}^{2} \right] d y \\ \\ \end{gathered} [/tex]
[tex]\begin{gathered}\rm\implies \:\sf \: \displaystyle \sf \dfrac{dy}{dx} = \dfrac{x y \log \left|\dfrac{y}{x}\right| }{\left[x^{2} \log \left|\dfrac{y}{x}\right| \: - \: {y}^{2} \right]} \\ \\ \end{gathered}[/tex]
Let assume that
[tex]\begin{gathered}\rm\implies \:\sf \:f(x, \: y)= \dfrac{x y \log \left|\dfrac{y}{x}\right| }{\left[x^{2} \log \left|\dfrac{y}{x}\right| \: - \: {y}^{2} \right]} \\ \\ \end{gathered}[/tex]
Now, Consider
[tex]\begin{gathered}\rm\implies \:\sf \:f(kx, \: ky)= \dfrac{(kx)(ky) \log \left|\dfrac{ky}{kx}\right| }{\left[ {k}^{2} x^{2} \log \left|\dfrac{ky}{kx}\right| \: - \: {k}^{2} {y}^{2} \right]} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\rm\implies \:\sf \:f(kx, \: ky)= \dfrac{ {k}^{2} \: x y \log \left|\dfrac{y}{x}\right| }{ {k}^{2} \: \left[x^{2} \log \left|\dfrac{y}{x}\right| \: - \: {y}^{2} \right]} \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\rm\implies \:\sf \:f(kx, \: ky)= \dfrac{ \: x y \log \left|\dfrac{y}{x}\right| }{ \: \left[x^{2} \log \left|\dfrac{y}{x}\right| \: - \: {y}^{2} \right]} \\ \\ \end{gathered}[/tex]
[tex] \begin{gathered}\rm\implies \:\sf \:f(kx, \: ky)= f(x, \: y) \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\rm\implies \:\sf \: f(x, \: y) \: is \: homogenous \: function.\\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\rm\implies \:\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x+\left[y^{2}-x^{2} \log \left|\frac{y}{x}\right|\right] d y=0 \: is \: homogenous\\ \\ \end{gathered}[/tex]
Alternative Method :-
Given differential equation is
[tex]\begin{gathered}\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x+\left[y^{2}-x^{2} \log \left|\frac{y}{x}\right|\right] d y=0 \\ \\ \end{gathered}[/tex]
can be rewritten as
[tex]\begin{gathered}\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x \: = \: - \: \left[y^{2}-x^{2} \log \left|\frac{y}{x}\right|\right] d y \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x \: = \: \left[x^{2} \log \left|\frac{y}{x}\right| \: - \: {y}^{2} \right] d y \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\sf \: \displaystyle \sf \dfrac{xy}{ {x}^{2} } \log \left|\frac{y}{x}\right| d x \: = \: \left[ \frac{ {x}^{2} }{ {x}^{2} } \log \left|\frac{y}{x}\right| \: - \: \frac{ {y}^{2} }{ {x}^{2} } \right] d y \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\sf \: \displaystyle \sf \dfrac{y}{ x} \log \left|\frac{y}{x}\right| d x \: = \: \left[ \log \left|\frac{y}{x}\right| \: - \: \frac{ {y}^{2} }{ {x}^{2} } \right] d y \\ \\ \end{gathered}[/tex]
[tex]\begin{gathered}\sf \: \dfrac{dy}{dx} \: = \: \dfrac{\displaystyle \sf \dfrac{y}{ x} \log \left|\frac{y}{x}\right|}{\left[\log \left|\dfrac{y}{x}\right| \: - \: \dfrac{ {y}^{2} }{ {x}^{2} } \right] } \: \: \\ \\ \end{gathered}[/tex]
which is of the form
[tex]\begin{gathered} \blue{\sf \: \dfrac{dy}{dx} \: = \: {x}^{n} \: f\bigg(\dfrac{y}{x} \bigg) } \: \: \\ \\ \end{gathered}[/tex]
Hence,
[tex]\begin{gathered}\rm\implies \:\sf \: \displaystyle \sf x y \log \left|\frac{y}{x}\right| d x+\left[y^{2}-x^{2} \log \left|\frac{y}{x}\right|\right] d y=0 \: is \: homogenous\\ \\ \end{gathered}[/tex]
Answer:
Hence,
⟹ xylog ∣∣∣∣y/x∣∣∣∣ dx + [y² − x² log ∣∣∣∣y/x∣∣∣∣ ] dy = 0 is homogeneous.
[tex]thanks[/tex]