Question

If ${\text{u = }}\log \left( {{{\text{x}}^{\text{2}}}{\text{ + }}{{\text{y}}^{\text{2}}}{\text{ + }}{{\text{z}}^{\text{2}}}} \right)$, verify $\dfrac{{{\partial ^{\text{2}}}{\text{u}}}}{{\partial {\text{x}}\partial {\text{y}}}}{\text{ = }}\dfrac{{{\partial ^{\text{2}}}{\text{u}}}}{{\partial {\text{x}}\partial {\text{y}}}}$

Answer 1

In this type of problem you need to verify the equation by using differentiation methods.
The key point in these questions is to check whether both sides are equal.
By using differentiation methods, given equations be differentiating with respect to x and y.
The first method to find the value of L.H.S, we are going to solve the given equation by differentiation method and differentiate the equation with respect to ${\text{x}}$ and ${\text{y}}$.
Now, substitute the differentiation values in the used formula then we get L.H.S.
Next R.H.S, we are going to solve the given equation by differentiation method and differentiate the equation with respect to ${\text{x}}$ and ${\text{y}}$.
Now, substitute the differentiation values in the used formula then we get R.H.S. We have to find the $\dfrac{{{\partial ^{\text{2}}}{\text{u}}}}{{\partial {\text{x}}\partial {\text{y}}}}{\text{ = }}\dfrac{{{\partial ^{\text{2}}}{\text{u}}}}{{\partial {\text{x}}\partial {\text{y}}}}$. For that we are going to solve using differentiation methods. And also we are going to substitute the values that have been given in the complete step-by-step solution.

Formula used: $\dfrac{{{\partial ^2}{\text{u}}}}{{\partial {\text{x }}\partial {\text{y}}}} = \dfrac{\partial }{{\partial {\text{x}}}}\left( {\dfrac{{\partial {\text{u}}}}{{\partial {\text{y}}}}} \right)$
Where ${\text{u}}$ is constant
${\text{log x = }}\dfrac{1}{{\text{x}}}$
Where ${\text{x}}$ is constant

Complete step-by-step solution:
Here it is a given that expression is $\dfrac{{{\partial ^{\text{2}}}{\text{u}}}}{{\partial {\text{x}}\partial {\text{y}}}}{\text{ = }}\dfrac{{{\partial ^{\text{2}}}{\text{u}}}}{{\partial {\text{x}}\partial {\text{y}}}}$. We have to verify the value of the given expression.
They give a differentiation equation to solve this express. So by using the differentiation method we are going to find the value of given expression.
Given ${\text{u = }}\log \left( {{{\text{x}}^{\text{2}}}{\text{ + }}{{\text{y}}^{\text{2}}}{\text{ + }}{{\text{z}}^{\text{2}}}} \right)$
Differentiate ${\text{u}}$ with respect to x,
$\dfrac{{\partial {\text{u}}}}{{\partial {\text{x}}}}{\text{ }} = {\text{ }}\dfrac{{2{\text{x}}}}{{{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}}}{\text{ }}$
Consider the given value to differentiate ${\text{u}}$ with respect to ${\text{y}}$,
$\dfrac{{\partial {\text{u}}}}{{\partial {\text{y}}}}{\text{ = }}\dfrac{{2{\text{y}}}}{{{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}}}$
By using formula of differentiation, we get
Now, $\dfrac{{{\partial ^2}{\text{u}}}}{{\partial {\text{y }}\partial {\text{x}}}} = \dfrac{\partial }{{\partial {\text{y}}}}\left( {\dfrac{{\partial {\text{u}}}}{{\partial {\text{x}}}}} \right)$
Substitute the values of differentiation $\dfrac{{\partial {\text{u}}}}{{\partial {\text{x}}}}$ and $\dfrac{{\partial {\text{u}}}}{{\partial {\text{y}}}}$, we have
$\dfrac{{{\partial ^2}{\text{u}}}}{{\partial {\text{y}}\partial {\text{x}}}}{\text{ = }}\dfrac{{\left( {{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}} \right)\dfrac{\partial }{{\partial {\text{y}}}}\left( {2{\text{x}}} \right) - 2{\text{x}}\dfrac{\partial }{{\partial {\text{y}}}}\left( {{{\text{x}}^2} + {{\text{y}}^2} + {z^2}} \right)}}{{{{\left( {{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}} \right)}^2}}}$
Differentiate the terms, we get
$\Rightarrow \dfrac{{0 - 4{\text{xy}}}}{{{{\left( {{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}} \right)}^2}}}$
$\Rightarrow \dfrac{{ - 4{\text{xy}}}}{{{{\left( {{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}} \right)}^2}}} - - - - - - \left( 1 \right)$
By using formula of differentiation,
$\dfrac{{{\partial ^2}{\text{u}}}}{{\partial {\text{x }}\partial {\text{y}}}} = \dfrac{\partial }{{\partial {\text{x}}}}\left( {\dfrac{{\partial {\text{u}}}}{{\partial {\text{y}}}}} \right)$
Substitute the values of $\dfrac{{\partial {\text{u}}}}{{\partial {\text{x}}}}{\text{ and }}\dfrac{{\partial {\text{u}}}}{{\partial {\text{y}}}}$, we have
$\dfrac{{{\partial ^2}{\text{u}}}}{{\partial {\text{y}}\partial {\text{x}}}}{\text{ = }}\dfrac{{\left( {{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}} \right)\dfrac{\partial }{{\partial {\text{x}}}}\left( {2{\text{y}}} \right) - 2{\text{y}}\dfrac{\partial }{{\partial {\text{x}}}}\left( {{{\text{x}}^2} + {{\text{y}}^2} + {z^2}} \right)}}{{{{\left( {{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}} \right)}^2}}}$
Differentiate the terms, we get
$\Rightarrow \dfrac{{0 - 4{\text{xy}}}}{{{{\left( {{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}} \right)}^2}}}$
$\Rightarrow \dfrac{{ - 4{\text{xy}}}}{{{{\left( {{{\text{x}}^2} + {{\text{y}}^2} + {{\text{z}}^2}} \right)}^2}}} - - - - - - \left( 2 \right)$
From $\left( {\text{1}} \right)$ and $\left( {\text{2}} \right)$ we have,
$\dfrac{{{\partial ^{\text{2}}}{\text{u}}}}{{\partial {\text{x}}\partial {\text{y}}}}{\text{ }}$ ${\text{ = }}\dfrac{{{\partial ^{\text{2}}}{\text{u}}}}{{\partial {\text{x}}\partial {\text{y}}}}$
Hence we verified that $\dfrac{{{\partial ^{\text{2}}}{\text{u}}}}{{\partial {\text{x}}\partial {\text{y}}}}{\text{ = }}\dfrac{{{\partial ^{\text{2}}}{\text{u}}}}{{\partial {\text{x}}\partial {\text{y}}}}$.

Hence Verified.

Note: Partial differential equations are ubiquitous in mathematically-oriented scientific fields, such as physics and engineering. For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, general relativity and quantum mechanics. They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations.