Formation of Ordinary Differential Equations: With General Solutions, Q&A

In most physical phenomena, we can observe the process but cannot directly work out the differential equation that is at work. As a result, we have the general solution at our disposal before we know the equation of which it is the solution. Let’s begin with the topic to understand the ordinary differential equations in further more detail.

Application of Ordinary Differential Equations

In Mechanics, it was experimentally observed that the velocity of a freely falling body, initially at rest, increases at a rate directly proportional to the square root of vertical distance it covers. It can be stated mathematically as \(v = \sqrt{2gh}\) since we actually know the constant of proportionality here.

ordinary differential equations

Now following this, one can differentiate the expression with respect to time to get the relation between the body’s acceleration and velocity, or directly get a differential equation in v and h, according to the need of the hour.

The resolution of the situation described above is the task at hand here. We will be given a general solution with n arbitrary constants, and we will have to find the nth order differential equation satisfied by the solution. For a clear and direct approach, we shall use the algorithm described below to find the required ordinary differential equations for all types of problems.

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The Algorithm

Given: A general solution with n arbitrary constants. The steps necessary to find the ordinary differential equations satisfied by this solution are –

Differentiate the general solution with respect to the independent variable exactly n times.
Use the (n+1) number of expressions (n derivatives) obtained to eliminate the n arbitrary constants in terms of the dependent variable or its derivatives.
Obtain the final expression which contains absolutely no arbitrary constant. This is the required differential equation.

Now go through the solved example below to get a proper idea about the working of this method.

Solved Examples For You

Question 1: Find the differential equation of the two-parameter family of conics \(ax^2 + by^2 = 1\), where a and b are arbitrary constants.

Answer : Here, we will be dealing with 3 expressions to eliminate the 2 arbitrary constants. One expression is the general solution of the differential equation, already given to us. The other two expressions are found by differentiating the general solution in a similar manner.

The above approach works theoretically, but is cumbersome and could lead to errors. A better approach to obtain the differential equation is to differentiate the given general solution and eliminate one of the arbitrary constants at every step, using the known expressions.

In the end, we are left with an expression containing no arbitrary constants, which would be our required solution.
This method is worked upon below –

\(ax^2 + by^2 = 1\) …. (1)

Differentiating with respect to x –

\(2ax + 2byy’ = 0\)

\(a = -\frac{byy’}{x}\) …. (2)

Substituting the value of a from (2) in the expression (1) –

\((-\frac{byy’}{x})x^2 + by^2 = 1\)

\(-bxyy’ + by^2 = 1\)

This, as you may notice, has only 1 arbitrary constant. Let us differentiate this expression with respect to x –

\(-b[xyy” + xy’^2 + yy’] + 2byy’ = 0\)

Divide this equation by b (≠0) on both sides –

\(xyy” + xy’^2 – yy’ = 0\)

\(xy\frac{d^2y}{dx^2} + x(\frac{dy}{dx})^2 – y\frac{dy}{dx} = 0\)

which is the required differential equation of order = 2. This brings us to the end of this topic. You must practice a lot of similar problems to properly understand how to eliminate the arbitrary constants most efficiently to get the resultant differential equation for a given general solution.

Question 2: Explain the order of differential equation?

Answer: Order of a differential equation refers to the order of the highest derivative. Experts also call it as differential coefficient that exists in the equation.

Question 3: What is meant by the first order of differential equation?

Answer: A first-order differential equation is one in which ƒ(x, y) happens to be a function of two variables defined on a xy-plane’s region. The equation belongs to the first order due to the fact that it involves only the first derivative dy dx.

Question 4: What are the various types of differential equations?

Answer: One can arrange all differential equations into two major types. These two types of the differential equation are partial differential equation and ordinary differential equation. A partial differential equation refers to a differential equation that deals with partial derivatives. An ordinary differential equation, in contrast, refers to a differential equation that does not involve partial derivatives.

Question 5: Explain the second order differential equation?

Answer: A second order differential equation refers to a type of equation such that it deals with the unknown function y, its derivatives y’ and y”. Furthermore, this equation also deals with the variable x.