Problem set in DE
Note: Classify the following as linear or non-linear, and determine the order
1). (1 − 𝑥)𝑦 ′′ − 4𝑥𝑦 ′ + 5𝑦 = cos 𝑥
Answer: (linear, 2nd order) why?
. Why linear? since as we can see that the dependent variable y’’ has a 1st degree we can conclude that
its linear
. Why it is 2nd order? Since the highest derivative is 2 and its free from radicals, we can conclude that it’s
in 2nd order
2). 𝜒
𝑑3𝑦
𝑑𝑥 3
𝑑𝑦
𝑑𝑥
−( ) 4 +𝑦 = 0
Answer: (Non-linear, 3rd order) why?
. Why its non-linear? Since a linear equation only has a 1st degree to be called a linear in number 2 we
can see that the exponent of the derivative (dy/dx) is 4 we can conclude that its non-linear
. Why its in 3rd order? Since the highest derivative is 3 and it free from radicals, we can conclude that its
in 3rd order
𝑑2 𝑢
𝑑𝑢
3. 𝑑𝑟2 + 𝑑𝑟 + 𝑢 = cos⁡(𝑟+u)
Answer: (Non-linear, 2nd order) why?
. Why its non-linear its non-linear because there is non-linear function of u or its derivatives.
. Why it is 2nd order? Since the highest derivative is 2 and its free from radicals, we can conclude that it’s
in 2nd order
4.
𝑑2 𝑦
𝑑𝑥 2
𝑑𝑦
𝑑𝑥
= √1 + ( ) 2
Answer: (Non-linear, 2nd order) why?
First squared both side
2
2
𝑑2 𝑦
𝑑𝑦 2
( 2 ) = (√1 + ( ) )
𝑑𝑥
𝑑𝑥
𝑑2 𝑦
2
𝑑𝑦 2
(𝑑𝑥 2 ) = 1 + (𝑑𝑥 ) obviously its non-linear because base on a condition (The dependent variable y, and
all its derivatives y 0 , y 00 , ..., y (n) , are of first degree (i.e. the power of these terms involving y is 1) so
as we have raise to 2 here in the derivative we can conclude that its non-linear
. Why it is 2nd order? Since the highest derivative is 2 and its free from radicals, we can conclude that it’s
in 2nd order