Question

Write the order and degree of the differential equation \[y = x\dfrac{{dy}}{{dx}} + a\sqrt {1 + {{\left( {\dfrac{{dy}}{{dx}}} \right)}^2}} \]

Answer 1

Hint: We use the concept of order and degree of a differential equation and apply them on the given equations to find the answer. Shift all the values except square root one side of the equation and use the formula of \[{(a - b)^2} = {a^2} + {b^2} - 2ab\] to solve the required side.

Complete step-by-step answer:
We have to find the order and degree of the differential equation\[y = x\dfrac{{dy}}{{dx}} + a\sqrt {1 + {{\left( {\dfrac{{dy}}{{dx}}} \right)}^2}} \] ……………….... (1)
Shift the term\[x\dfrac{{dy}}{{dx}}\] from RHS to LHS of the equation
\[ \Rightarrow y - x\dfrac{{dy}}{{dx}} = a\sqrt {1 + {{\left( {\dfrac{{dy}}{{dx}}} \right)}^2}} \]
Square both sides of the equation
\[ \Rightarrow {\left( {y - x\dfrac{{dy}}{{dx}}} \right)^2} = {\left( {a\sqrt {1 + {{\left( {\dfrac{{dy}}{{dx}}} \right)}^2}} } \right)^2}\]
Use the formula \[{(a - b)^2} = {a^2} + {b^2} - 2ab\]to expand LHS
\[ \Rightarrow {\left( y \right)^2} + {\left( {x\dfrac{{dy}}{{dx}}} \right)^2} - 2\left( y \right)\left( {x\dfrac{{dy}}{{dx}}} \right) = {a^2}{\left( {\sqrt {1 + {{\left( {\dfrac{{dy}}{{dx}}} \right)}^2}} } \right)^2}\]
\[ \Rightarrow {y^2} + {x^2}{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - 2xy\left( {\dfrac{{dy}}{{dx}}} \right) = {a^2}{\left( {\sqrt {1 + {{\left( {\dfrac{{dy}}{{dx}}} \right)}^2}} } \right)^2}\]
Cancel square root by square power on RHS of the equation.
\[ \Rightarrow {y^2} + {x^2}{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - 2xy\left( {\dfrac{{dy}}{{dx}}} \right) = {a^2}\left( {1 + {{\left( {\dfrac{{dy}}{{dx}}} \right)}^2}} \right)\]
\[ \Rightarrow {y^2} + {x^2}{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - 2xy\left( {\dfrac{{dy}}{{dx}}} \right) = {a^2} + {a^2}{\left( {\dfrac{{dy}}{{dx}}} \right)^2}\]
Shift all values to LHS of the equation
\[ \Rightarrow {y^2} + {x^2}{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - 2xy\left( {\dfrac{{dy}}{{dx}}} \right) - {a^2} - {a^2}{\left( {\dfrac{{dy}}{{dx}}} \right)^2} = 0\]
Collect terms with same variables
\[ \Rightarrow \left\{ {{x^2}{{\left( {\dfrac{{dy}}{{dx}}} \right)}^2} - {a^2}{{\left( {\dfrac{{dy}}{{dx}}} \right)}^2}} \right\} - 2xy\left( {\dfrac{{dy}}{{dx}}} \right) + {y^2} - {a^2} = 0\]
\[ \Rightarrow \left\{ {{x^2} - {a^2}} \right\}{\left( {\dfrac{{dy}}{{dx}}} \right)^2} - 2xy\left( {\dfrac{{dy}}{{dx}}} \right) + {y^2} - {a^2} = 0\]...................… (2)
Now we use the definition of order and degree on equation (2)
Since, highest order derivative is \[\dfrac{{dy}}{{dx}}\]
\[ \Rightarrow \]Order \[ = 1\]
Also, the highest power of the highest order derivative is given by \[{\left( {\dfrac{{dy}}{{dx}}} \right)^2}\]
\[ \Rightarrow \]Degree \[ = 2\]

\[\therefore \]Order of the differential equation is 1 and the degree of the differential equation is 2.

Note: * Order of a differential equation is the order of the highest derivative. Example: In a differential equation \[\dfrac{{{d^3}y}}{{d{x^3}}} - 3\dfrac{{{d^2}y}}{{d{x^2}}} + \dfrac{{dy}}{{dx}} - 2y = 3\]has order 3, as the highest derivative of third order.
* Degree of a differential equation is the highest power of the highest order derivative. Example: In a differential equation\[{\left( {\dfrac{{{d^3}y}}{{d{x^3}}}} \right)^2} - 3\dfrac{{{d^2}y}}{{d{x^2}}} + {\left( {\dfrac{{dy}}{{dx}}} \right)^4} - 2y = 3\] has order 3 and degree 2, as the power of the highest order is 2.