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Find the number of arbitrary constants in the general solution of differential equations of order 3.
A. 0
B. 2
C. 3
D. 4

Last updated date: 23rd Jun 2024
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Hint: We will use the fact that we will have as much constants as the order of the equation. By using this fact, we will get the answer of 3 arbitrary constants in the answer and thus C) 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 from.
When we have an equation of degree 3 with us, we can integrate it on both the sides.
By integrating it, we will get one constant in the new equation with degree 2.
Now, if we integrate it again, we will get one more constant and new equation with degree 1.
Now, 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.
Take $\dfrac{{{d^3}y}}{{d{x^3}}} = 1$.
Integrating this on both sides, we get: $\dfrac{{{d^2}y}}{{d{x^2}}} = x + a$, where a is the first arbitrary constant.
Integrating again, we will get: $\dfrac{{dy}}{{dx}} = \dfrac{{{x^2}}}{2} + ax + b$, where b is the second arbitrary constant.
Integrating for the last time, we have: $y = \dfrac{{{x^3}}}{6} + \dfrac{{a{x^2}}}{2} + bx + c$, where c is the third arbitrary constant.
Hence, we get three arbitrary constants for third order equations.
Hence, we can use this fact now.
Hence, the answer will be 3

So, the correct answer is “Option C”.

Note: The students might make the mistake if the question had the word “particular solution” instead of general solution because in particular solution, we have 0 arbitrary constants. Always do remember the difference.
Never do any solution with the use of an example. Here, we saw an example to understand the view point better, not to prove our fact.