Check Online MTH401 Assignment No: 1 Fall 2014 Solution Due date 01/12/2014
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Distinguish the linear and non-linear differential equations out of the following.
An equation is a 1st order linear equation if it can be written as dy / dx + p(x)y = q(x).
(i). x dy / dx + y = 2x.
There is no function p(x) in the above equation hence it is non linear.
(ii). xy dy / dx + (x^2)(y^2) = 2.
Here y is raise to the power 2, for linear eq y has to be power of 1. hence it is also on linear.
(iii). dy / dx + 3(x^2)y = 6x^2.
it can be written as dy / dx + p(x)y = q(x).
This equation is linear.
(iv). cos^3x dy / dx + ycosx = sinx
it can be written as dy / dx + p(x)y = q(x).
It is linear.
(v). x^2dy + y^2x = 0
I thinks its non linear.
Errors and omissions are excepted.
Q1). Linear and non linear equations.
An equation is a 1st order linear equation if it can be written as dy / dx + p(x)y = q(x).
(i). x dy / dx + y = 2x.
There is no function p(x) in the above equation hence it is non linear.
(ii). xy dy / dx + (x^2)(y^2) = 2.
Here y is raise to the power 2, for linear eq y has to be power of 1. hence it is also on linear.
(iii). dy / dx + 3(x^2)y = 6x^2.
it can be written as dy / dx + p(x)y = q(x).
This equation is linear.
(iv). cos^3x dy / dx + ycosx = sinx
it can be written as dy / dx + p(x)y = q(x).
It is linear.
(v). x^2dy + y^2x = 0
I thinks its non linear.
Errors and omissions are excepted.
In a differential equation, when the variables and their derivatives are only multiplied by constants, then the equation is linear. The variables and their derivatives must always appear as a simple first power. Here are some examples.
x'' + x = 0 is linear
x'' + 2x' + x = 0 is linear
x' + 1/x = 0 is non-linear because 1/x is not a first power
x' + x2 = 0 is non-linear because x2 is not a first power
x'' + sin(x) = 0 is non-linear because sin(x) is not a first power
x x' = 1 is non-linear because x' is not multiplied by a constant.
Similar rules apply to multiple variable problems.
x' + y' = 0 is linear
x y' = 1 is non-linear because y' is not multiplied by a constant
Note, however, that an exception is made for the time variable t (ie. the variable that we are differentiating by). We can have any function of t appear in the equation, but still have an equation that is linear in x.
x'' + 2 x' + x = sin(t) is linear in x
x' + t2x = 0 is linear in x
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