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9. differential equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths9. differential equations1 Mark
MCQ
Which one of the following functions is not homogeneous?
  • A
    $f (x, y) =\frac{{x - y}}{{{x^2} + {y^2}}}$
  • B
    $f (x, y) =  {x^{\frac{1}{3}}}\cdot{y^{ - \,\,\frac{2}{3}}}{\tan ^{ - 1}}\frac{x}{y}$
  • C
    $f (x, y) = x (ln  \sqrt {{x^2} + {y^2}}  \, - ln y)+ye^{x/y }$
  • ✓
    $f(x,y)=x \ \left[ {\ln \frac{{2{x^2} + {y^2}}}{x} - \ln (x + y)} \right] \, + \,y^2 \tan \frac{{x + 2y}}{{3x - y}}$
✓

Answer

Correct option: D.
$f(x,y)=x \ \left[ {\ln \frac{{2{x^2} + {y^2}}}{x} - \ln (x + y)} \right] \, + \,y^2 \tan \frac{{x + 2y}}{{3x - y}}$
d
$(A)$  $f (\lambda x, \lambda y) =  \frac{{\lambda (x - y)}}{{{\lambda ^2}({x^3} + {y^2})}} = \lambda ^{-1} f (x, y) \Rightarrow$ homogeneous of degree $(-1).$
$(B)$  $f (\lambda x, \lambda y) =  {(\lambda x)^{\frac{1}{3}}}{(\lambda y)^{\frac{{ - 2}}{3}}}{\tan ^{ - 1}}\frac{x}{y} = {\lambda ^{\frac{{ - 1}}{3}}}{x^{\frac{1}{3}}}{y^{\frac{{ - 2}}{3}}}{\tan ^{ - 1}}\frac{x}{y}$
$= {\lambda ^{\frac{{ - 1}}{3}}} \,f (x, y) \Rightarrow$ homogeneous
$(C)$  $f (\lambda x, \lambda y) = \lambda x \left( {\ln \sqrt {{\lambda ^2}({x^2} + {y^2})}  - \ln \,\lambda y} \right) +\, \lambda y{e^{\frac{x}{y}}}$
$= \lambda x  \, \left( {\ln \left( {\frac{{\lambda \sqrt {({x^2} + {y^2})} }}{{\lambda y}}} \right)} \right) \,+ \,\lambda y{e^{\frac{x}{y}}}$
$= \lambda  \left[ {x\,\,\left( {\ln \sqrt {{x^2} + {y^2}}  - \ln \,y} \right) + y{e^{\frac{x}{y}}}} \right]$
$= \lambda f (x, y) \Rightarrow$ homogeneous.
$(D)$  $f (\lambda x, \lambda y) = \lambda x \, \left[ {\ln \frac{{2{\lambda ^2}{x^2} + {\lambda ^2}{y^2}}}{{\lambda x\,\cdot\,\lambda (x + y)}}} \right] \, + \lambda ^2x^2 \tan \frac{{x + 2y}}{{3x - y}}$
$= \lambda x   \left[ {\ln \frac{{2{x^2} + {y^2}}}{{x(x + y) }}} \right] \, + \lambda ^2x^2 tan \Rightarrow$ non homogeneous

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