Question

The number of arbitrary constants in the general solution of a differential equation of fourth order are
A) 0
B) 2
C) 3
D) 4

Answer 1

Hint: We will use the fact that we will have as many constants as the order of the equation. Take an example of fourth order and integrate it four times. Every time a new arbitrary constant will come. By using this fact, we will get the answer of 4 arbitrary constants in the answer and thus (D) being the answer.

Complete step by step answer:
We will use the fact that we will have as many constants as the order of the equation.
Before using this fact, let us get to know where this arises.
When we have an equation of degree 4 with us, we can integrate it on both sides.
Now, if we integrate it again, we will get a one more constant and new equation with degree 2.
Now, we have to repeat the integration which will lead us to 3 arbitrary constants in all.
Again, we have to repeat the integration which will lead us to 3 arbitrary constants in all and we will get the equation of the curve finally.
Let us see an example to get a clearer picture.
$\dfrac{{{d^4}y}}{{d{x^4}}} = 1$
Integrate on both sides,
$ \Rightarrow \dfrac{{{d^3}y}}{{d{x^3}}} = x + a$
where $a$ is the first arbitrary constant.
Integrate on both sides again,
$ \Rightarrow \dfrac{{{d^2}y}}{{d{x^2}}} = \dfrac{{{x^2}}}{2} + ax + b$
where $b$ is the second arbitrary constant.
Integrate on both sides one more time,
$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{{x^3}}}{6} + \dfrac{{a{x^2}}}{2} + bx + c$
where $c$ is the third arbitrary constant.
Integrate on both sides last time,
$ \Rightarrow y = \dfrac{{{x^4}}}{{24}} + \dfrac{{a{x^3}}}{6} + \dfrac{{b{x^2}}}{2} + cx + d$
where $d$ is the fourth arbitrary constant.
So, the answer will be 4.

Hence, option (D) is the correct answer.

Note:
The students might make the mistake if the question had the word “particular solution” instead of the general solution because, in a particular solution, we have 0 arbitrary constants. Always do remember the difference.
General solution is a solution of a differential equation which contains arbitrary constants equal to order of differential equation.
Particular solution is a solution obtained from a general solution, by assigning some values to arbitrary constants.