Question

Which one of the following differential equations is not linear?
A) \[\dfrac{{{d^2}y}}{{d{x^2}}} + 4y = 0\]
B) \[x\dfrac{{dy}}{{dx}} + y = {x^3}\]
C) \[{\left( {x - y} \right)^{^2}}\dfrac{{dy}}{{dx}} = 9\]
D) \[{\cos ^2}x\dfrac{{dy}}{{dx}} + y = \tan x\]

Answer 1

Hint: A linear differential equation is an equation in which the degree as well as the order of the differential equation is one i.e. the highest power and the order of the derivative of the variable should be one. Hence we will check each of the options one by one to get the desired answer.

Complete step-by-step answer:
We know that a linear differential equation is an equation in which the degree as well as the order of the differential equation is one.
Let us check for each of the options whether they are linear or not.
A) \[\dfrac{{{d^2}y}}{{d{x^2}}} + 4y = 0\]
Here the degree of the derivative of x is one but its order is 2
Hence A is not a linear differential equation.
B) \[x\dfrac{{dy}}{{dx}} + y = {x^3}\]
Here the degree of the derivative of x is one as well as its order is 1
Hence B is a linear differential equation.
C) \[{\left( {x - y} \right)^{^2}}\dfrac{{dy}}{{dx}} = 9\]
Here the degree of the derivative of x is one as well as its order is 1
Hence C is a linear differential equation.
D) \[{\cos ^2}x\dfrac{{dy}}{{dx}} + y = \tan x\]
Here the degree of the derivative of x is one as well as its order is 1
Hence D is a linear differential equation.

Therefore, option A is the correct option.

Additional Information:
Order of a differential equation is the order of the highest derivative of the variable in the differential equation.
Eg- the order of the following differential equation is 3.
\[\dfrac{{{d^3}y}}{{d{x^3}}} + x = 0\]
The degree of the differential equation is the highest power to which the derivative of the variable is raised.
Eg – the degree of the following differential equation is 2.
\[{\left( {\dfrac{{dy}}{{dx}}} \right)^2} + y = 0\]

Note: Students should keep in mind that they only have to observe the degree and order of the derivative and not the variable itself.
A non linear differential equation may have degree or order of the derivative as 2 or more than 2.