Order Of A Differential Equation

Calculus Level 3

What is the order of the differential equation $x^2.\left( \dfrac{d^2y}{dx^2} \right)^6+y^{\frac{-2}{3}}.\sqrt{1+\left( \dfrac{d^3y}{dx^3} \right)^5}+\dfrac{d^2}{dx^2}. \left( \dfrac{d^2y}{dx^2} \right)^{\frac{-2}{3}} =0?$

Not defined 6 3 4

1 solution

Ayush Verma
Dec 10, 2014

Order of darivative of first term is 2 & that of second term is 3.

For third term,

$\cfrac { { d }^{ 2 } }{ { dx }^{ 2 } } .{ \left( \cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) }^{ \cfrac { -2 }{ 3 } }=\cfrac { d }{ dx } \left\{ \cfrac { d }{ dx } { \left( \cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) }^{ \cfrac { -2 }{ 3 } } \right\} \\ \\ \quad =\cfrac { d }{ dx } \left\{ \cfrac { -2 }{ 3 } { \left( \cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) }^{ \cfrac { -5 }{ 3 } }.\cfrac { { d }^{ 3 }y }{ d{ x }^{ 3 } } \right\} \\ \\ \quad =\cfrac { { d }^{ 3 }y }{ d{ x }^{ 3 } } .\cfrac { d }{ dx } \left\{ \cfrac { -2 }{ 3 } { \left( \cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) }^{ \cfrac { -5 }{ 3 } } \right\} +\cfrac { -2 }{ 3 } { \left( \cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } \right) }^{ \cfrac { -5 }{ 3 } }.\cfrac { { d }^{ 4 }y }{ d{ x }^{ 4 } } \\ \\ Clearly\quad order\quad of\quad this\quad term\quad is\quad 4$

So,order of the highest order derivative present in the equation is 4 .

Order Of A Differential Equation

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