Exercise 8.7 - Chapter 8 Differentials and Partial Derivatives 12th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $8.7$
Question 1.
In each of the following cases, determine whether the following function is homogeneous or not. If it is so, find the degree.
(i) $f(x, y)=x^2 y+6 x^3+7$
(ii) $h(x, y)=\frac{6 x^2 y^3-\pi y^5+9 x^4 y}{2020 x^2+2019 y^2}$
(iii) $g(x, y, z)=\frac{\sqrt{3 x^2+5 y^2+z^2}}{4 x+7 y}$
(iv) $\mathrm{U}(x, y, z)=x y+\sin \left(\frac{y^2-2 z^2}{x y}\right)$
Solution:
(i)
$
\begin{aligned}
f(x, y) & =x^2 y+6 x^3+7 \\
f(t x, t y) & =(t x)^2(t y)+6(t x)^3+7 \\
f(t x, t y) & =t^3\left(x^2 y\right)+t^3\left(6 x^3\right)+7
\end{aligned}
$
It is not a homogeneous function
(ii)
$
\begin{aligned}
h(x, y) & =\frac{6 x^2 y^3-\pi y^5+9 x^4 y}{2020 x^2+2019 y^2} \\
h(t x, t y) & =\frac{6(t x)^2(t y)^3-\pi(t y)^5+9(t x)^4(t y)}{2020(t x)^2+2019(t y)^2} \\
h(t x, t y) & =\frac{t^5\left[6 x^2 y^3-\pi y^5+9 x^4 y\right]}{t^2\left[2020 x^2+2019 y^2\right]} \\
h(t x, t y) & =t^3 h(x, y)
\end{aligned}
$
$\therefore$ It is a homogeneous function with degree 3 .
(iii) $\quad g(x, y, z)=\frac{\sqrt{3 x^2+5 y^2+z^2}}{4 x+7 y}$

$
\begin{aligned}
g(t x, t y, t z) & =\frac{\sqrt{3 t^2 x^2+5 t^2 y^2+t^2 z^2}}{4 t x+7 t y} \\
& =\frac{\sqrt{t^2}}{t}\left[\frac{\sqrt{3 x^2+5 y^2+z^2}}{4 x+7 y}\right] \\
g(t x, t y, t z) & =t^0[g(x, y, z)]
\end{aligned}
$
$\therefore$ It is homogeneous function of degree 0 .
(iv)
$
\begin{aligned}
\mathrm{U}(x, y, z) & =x y+\sin \left(\frac{y^2-2 z^2}{\partial y}\right) \\
\mathrm{U}(t x, t y, t z) & =(t x)(t y)+\sin \left(\frac{t^2 y^2-2 t^2 z^2}{(t x)(t y)}\right) \\
& =t^2(x y)+\sin \left(\frac{y^2-2 z^2}{x y}\right)
\end{aligned}
$
$\therefore$ It is not a homogeneous function

Question 2.
Prove that $f(x, y)=x^3-2 x^2 y+3 x y^2+y^3$ is homogeneous; what is the degree? Verify Euler's Theorem for $\mathrm{f}$.
Solution:
$
\begin{aligned}
& f(x, y)=x^3-2 x^2 y+3 x y^2+y^3 \\
& f(t x, t y)=t^3 x^3-2\left(t^2 x^2\right)(t y)+3(t x)\left(t^2 y^2\right)+t^3 y^3 \\
& =t^3\left[x^3-2 x^2 y+3 x y^2+y^3\right] \\
& f(t x, t y)=t^3 f(x, y)
\end{aligned}
$
' $\mathrm{f}$ ' is a homogeneous function of degree 3. By Euler's theorem, we have
$
\begin{aligned}
x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y} & =3 f \\
\text { LHS } & =x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y} \\
& =x\left[3 x^2-4 x y+3 y^2\right]+y\left[-2 x^2+6 x y+3 y^2\right] \\
& =3 x^3-4 x^2 y+3 x y^2-2 x^2 y+6 x y^2+3 y^3 \\
& =3 x^3-6 x^2 y+9 x y^2+3 y^3=3\left[x^3-2 x^2 y+3 x y^2+y^3\right] \\
& =3 f=\text { RHS }
\end{aligned}
$
$\therefore$ Euler's Theorem verified

Question 3.
Prove that $\mathrm{g}(\mathrm{x}, \mathrm{y})=\mathrm{x} \log \left(\frac{y}{x}\right)$ is homogeneous; what is the degree? Verify Euler's Theorem for $\mathrm{g}$.

Solution:
$
\begin{aligned}
g(x, y) & =x \log \left(\frac{y}{x}\right) \\
g(t x, t y) & =(t x) \log \left(\frac{t y}{t x}\right) \\
& =t x \log \left(\frac{y}{x}\right) \\
g(t x, t y) & =t g(x, y)
\end{aligned}
$
$\therefore$ ' $g$ ' is a homogeneous function of degree 1. By Euler's theorem, we have
$
\begin{aligned}
& \begin{aligned}
x \frac{\partial g}{\partial x}+y \frac{\partial g}{\partial y} & =g \\
\text { LHS } & =x\left[x\left(\frac{x}{y} \cdot \frac{-y}{x^2}\right)+\log \left(\frac{y}{x}\right)\right]+y\left[x\left(\frac{x}{y} \cdot \frac{1}{x}\right)\right] \\
& =x\left[-1+\log \left(\frac{y}{x}\right)\right]+x \\
& =-x+x \log \left(\frac{y}{x}\right)+x \\
& =x \log \left(\frac{y}{x}\right)=\text { RHS }
\end{aligned} \\
& \therefore \text { Euler's Theorem verified }
\end{aligned}
$

Question 4.
If $u(x, y)=\frac{x^2+y^2}{\sqrt{x+y}}$, prove that $x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=\frac{3}{2} u$
Solution:
$
\begin{aligned}
u(x, y) & =\frac{x^2+y^2}{\sqrt{x+y}} \\
u(t x, t y) & =\frac{t^2 x^2+t^2 y^2}{\sqrt{t x+t y}}=\frac{t^2}{\sqrt{t}}\left[\frac{x^2+y^2}{\sqrt{x+y}}\right] \\
u(t x, t y) & =t^{\frac{3}{2}} u(x, y)
\end{aligned}
$
' $u$ ' is a homogeneous function of degree $\frac{3}{2}$. By Euler's theorem, we have
$
x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=\frac{3}{2} u
$

Question 5.
If $v(x, y)=\log \left(\frac{x^2+y^2}{x+y}\right)$, prove that $x \frac{\partial v}{\partial x}+y \frac{\partial v}{\partial y}=1$
Solution:
$
v(x, y)=\log \left(\frac{x^2+y^2}{x+y}\right)
$
Let $f=e^v=\frac{x^2+y^2}{x+y}$
$\therefore$ ' $\mathrm{f}$ ' is a homogeneous function of degree 1. By Euler's theorem, we have
$
x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f
$
(but $f=e^v$ and $n=1$ )
$
\begin{gathered}
x \frac{\partial}{\partial x}\left(e^v\right)+y \frac{\partial}{\partial y}\left(e^v\right)=e^v \\
x e^v \frac{\partial v}{\partial x}+y e^v \frac{\partial v}{\partial y}=e^v \\
x \frac{\partial v}{\partial x}+y \frac{\partial v}{\partial y}=1
\end{gathered}
$

Question 6.
If $\boldsymbol{w}(\boldsymbol{x}, \boldsymbol{y}, z)=\log \left(\frac{5 x^3 y^4+7 y^2 x z^4-75 y^3 z^4}{x^2+y^2}\right)^2$, find $x \frac{\partial w}{\partial x}+y \frac{\partial w}{\partial y}+z \frac{\partial w}{\partial z}$
Solution:
$
w(x, y, z)=\log \left[\frac{5 x^3 y^4+7 y^2 x z^4-75 y^3 z^4}{x^2+y^2}\right]
$
Let $f=e^w=\frac{5 x^3 y^4+7 y^2 x z^4-75 y^3 z^4}{x^2+y^2}$
' $f$ ' is a homogeneous function of degree 5. By Euler's theorem, we have
$
x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}+z \frac{\partial f}{\partial z}=n f
$
(But $n=5 ; f=e^w$ )
$
\begin{aligned}
x \frac{\partial}{\partial x}\left(e^w\right)+y \frac{\partial}{\partial y}\left(e^w\right)+z \frac{\partial}{\partial z}\left(e^w\right) & =5 e^w \\
e^w\left[x \frac{\partial w}{\partial x}+y \frac{\partial w}{\partial y}+z \frac{\partial w}{\partial z}\right] & =5 e^w \\
x \frac{\partial w}{\partial x}+y \frac{\partial w}{\partial y}+z \frac{\partial w}{\partial z} & =5
\end{aligned}
$