Differential Equations

General and Particular Solutions of a Differential Equation

Differential Equations Solutions: A solution of a differential equation is a relation between the variables (independent and dependent), which is free of derivatives of any order, and which satisfies the differential equation identically. Now let’s get into the details of what ‘differential equations solutions’ actually are!

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Differential Equations Solutions

If we consider a general nth order differential equation –

\(F[x, y, \frac{dy}{dx}, ….. ,\frac{d^ny}{dx^n}] = 0\),

where F is a real function of its (n + 2) arguments – \( x, y, \frac{dy}{dx}, ….. ,\frac{d^ny}{dx^n}\).
Then a function f(x), defined in an interval x ∈ I and having an nth derivative (as well as all of the lower order derivatives) for all x ∈ I; is known as an explicit solution of the given differential equation only if –

\(F[x, f(x), f'(x),……f^{(n)}(x)] = 0\), for all x ∈ I.

A relation g(x,y) = 0, is known as the implicit solution of the given differential equation if it defines at least one real function f of the variable x on an interval I such that this function is an explicit solution of the differential equation on this interval, as per the above conditions.

Browse more Topics Under Differential Equations

General Solution of a Differential Equation

A General Solution of an nth order differential equation is one that involves n necessary arbitrary constants.

If we solve a first order differential equation by variables separable method, we necessarily have to introduce an arbitrary constant as soon as the integration is performed. Thus you can see that a solution of a differential equation of the first order has 1 necessary arbitrary constant after simplification.

Similarly, the general solution of a second order differential equation will contain 2 necessary arbitrary constants and so on. The general solution geometrically represents an n-parameter family of curves. For example, the general solution of the differential equation \(\frac{dy}{dx} = 3x^2\), which turns out to be \(y = x^3 + c\) where c is an arbitrary constant, denotes a one-parameter family of curves as shown in the figure below.

differential equations solutions

Particular Solution of a Differential Equation

A Particular Solution of a differential equation is a solution obtained from the General Solution by assigning specific values to the arbitrary constants. The conditions for calculating the values of the arbitrary constants can be provided to us in the form of an Initial-Value Problem, or Boundary Conditions, depending on the problem.

Singular Solution

The Singular Solution is also a Particular Solution of a given differential equation but it can’t be obtained from the General Solution by specifying the values of the arbitrary constants.

Solved Examples For You

Question 1: Determine whether the function \(f(t) = c_1e^t + c_2e^{-3t} + sint\) is a general solution of the differential equation given as –

\(\frac{d^2F}{dt^2}  + 2\frac{dF}{dt} – 3F = 2cost – 4sint\)

Also find the particular solution of the given differential equation satisfying the initial value conditions f(0) = 2 and f'(0) = -5.

Answer : The function f(t) must satisfy the differential equation in order to be a solution. So let us first write down the derivatives of f.

\(f(t) = c_1e^t + c_2e^{-3t} + sint\)

\(f'(t) = c_1e^t – 3c_2e^{-3t} + cost\)

\(f”(t) = c_1e^t + 9c_2e^{-3t} – sint\)

Now let us use these values for F in the left-hand side of the differential equation and compute the result.

\(\frac{d^2f}{dt^2}  + 2\frac{df}{dt} – 3f\)

\((c_1e^t + 9c_2e^{-3t} – sint) + 2(c_1e^t – 3c_2e^{-3t} + cost) -3(c_1e^t + c_2e^{-3t} + sint)\)

\(2cost – 4sint\), after cancellation of like terms.

Thus we find the left-hand side of the differential equation to be equal to the right-hand side after simplification. Therefore, given f(t) is a solution of the differential equation.

Besides, since the order of the differential equation = 2, and the number of arbitrary constants in the function f(t) = 2; we find that the solution given by f(t) is indeed the General Solution of the differential equation.

Determination of the Particular Solution –

From the expression \(f(t) = c_1e^t + c_2e^{-3t} + sint\), at t = 0 we get –

\(f(0) = c_1 + c_2 = 2\) ….. (1)

Similarly, from the expression \(f'(t) = c_1e^t – 3c_2e^{-3t} + cost\), at t = 0 we get –

\(f'(0) = c_1 – 3c_2 + 1 = -5\) ….. (2)

On solving the simultaneous linear equations (1) and (2), we can get the values of c1 and c2 as –

\(c_1 = 0\) and \(c_2 = 2\)

The required particular solution is –

\(f(t) = 2e^{-3t} + sint\)

Similarly, all other problems on diffferential equations solutions can be handled. This concludes our discussion on this topic of differential equations solutions.

Question 2: What is the importance of differential equations?

Answer: It is important as a technique for determining a function is that if we know the function and perhaps some of its derivatives at a specific point, then together with differential equation we can use this information to determine the function over its entire domain.

Question 3: Where are differential equations used?

Answer: These equations have an amazing capacity to forecast the world around us. In addition, they are used in a wide variety of disciplines, from biology, economics, chemistry, physics, and engineering. Moreover, they can define exponential growth and decay, the population growth of species or the change in investment return over time.

Question 4: Define differential equations?

Answer: It is an equation that relates one or more functions and their derivatives. Generally, we use the functions to signify physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.

Question 5: State the types of differential equations?

Answer: There are six types of differential equations. These are:

  1. Ordinary Differential Equations
  2. Partial Differential Equations
  3. Linear Differential Equations
  4. Non-linear Differential Equations
  5. Homogenous Differential Equations
  6. Non-homogenous Differential Equations

All these differential equations have different functions and use.

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