Solution of Differential Equation

A mathematical equation that connects a function to its derivatives is called a differential equation. These formulas can explain a wide variety of phenomena, including object motion and electrical circuit behaviour. In many disciplines, including physics, engineering, biology, and economics, they are indispensable tools.

The Differential equations can be classified based on the type of derivatives involved:

• Ordinary Differential Equations (ODEs): Involve derivatives with respect to a single independent variable.
• Partial Differential Equations (PDEs): Involve derivatives with respect to multiple independent variables.

Methods for Solving Differential Equations

To find the solution of differential equations, we can use the following methods:

• Separation of Variables
• Integrating Factor Method
• Homogeneous Equations
• Exact Differential Equations
• Numerical Methods

Separation of Variables

Separation of Variables is a method used to solve differential equations. It involves rewriting the differential equation in a form where the variables can be separated, allowing integration to be performed on each side independently.

This method is applicable to the separable equations where the equation can be written in the form:

\frac{dy}{dx} = g(x)h(y)

The variables can be separated and integrated to the find the solution.

Integrating Factor Method

Used for the first-order linear ODEs of the form:

\frac{dy}{dx} + P(x)y = Q(x)

The integrating factor is given by:

\mu(x) = e^{\int P(x)dx}

Homogeneous Equations

For homogeneous equations the equation can be rewritten by the substituting:

v = \frac{y}{x}reducing it to the separable form.

Exact Differential Equations

An equation of the form:

M(x, y) dx + N(x, y) dy = 0

is exact if:

\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}

The solution is obtained by the integrating the M with the respect to the x and N with the respect to y.

Numerical Methods

Numerical methods are essential for solving differential equations that cannot be solved analytically. These methods approximate the solutions using numerical techniques and are particularly useful for complex problems or those involving real-world data. Some of the common methods are:

• Euler's Method
• Runge-Kutta Method
• Backward Euler Method
• Finite Difference Method

General Solution of Differential Equations

The general solution of the differential equation encompasses all possible solutions and involves the arbitrary constants.

Applications of Differential Equations

The Differential equations are widely used in the various fields such as the physics, engineering, biology and more.

• Physics: The Modeling the motion of the objects, heat conduction and wave propagation.
• Biology: The Modeling population dynamics, spread of the diseases and chemical reactions.
• Engineering: The Analyzing the electrical circuits, control systems and mechanical vibrations.

Solved Examples

Example 1: Solve the differential equation: \frac{dy}{dx} = xy

Solution:

Separate the variables:

\frac{dy}{y} = x \, dx

Integrate both sides:

\ln|y| = \frac{x^2}{2} + C

Exponentiate both sides:

y = e^{\frac{x^2}{2} + C} = C'e^{\frac{x^2}{2}}

So, the solution is:

y = Ce^{\frac{x^2}{2}}

Example 2: Solve the differential equation: \frac{dy}{dx} + \frac{2y}{x} = x^2

Solution:

This is a first-order linear equation. The integrating factor is:

\mu(x) = e^{\int \frac{2}{x} \, dx} = x^2

The Multiply the entire equation by the x^2:

x^2 \frac{dy}{dx} + 2xy = x^4

This simplifies to:

\frac{d}{dx}(x^2 y) = x^4

The Integrate both sides:

x^2 y = \frac{x^5}{5} + C

So, the solution is:

y = \frac{x^3}{5} + \frac{C}{x^2}

Example 3: Solve the exact differential equation: (2xy + y²) dx + (x² + 2xy) dy = 0

Solution:

Check for exactness:

\frac{\partial M}{\partial y} = 2x + 2y \quad \text{and} \quad \frac{\partial N}{\partial x} = 2x + 2y

Since \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} the equation is exact.

Integrate M with the respect to x:

\int (2xy + y^2) \, dx = x^2 y + xy^2 + g(y)

Differentiate with the respect to y and compare with the N:

\frac{\partial}{\partial y}(x^2 y + xy^2 + g(y)) = x^2 + 2xy + g'(y)

So, g'(y) = 0 meaning g(y) is a constant. The solution is:

x ²y+xy ²=C

Example 4: Solve the second-order linear differential equation: y'' - 4y' + 4y = 0

Solution:

The characteristic equation is:

r² - 4r + 4 = 0

This factors as:

(r - 2)²= 0

So, r = 2 is a repeated root. The general solution is:

y(x) = (C_1 + C_2 x) e^{2x}

Example 5: Solve the differential equation:\frac{dy}{dx} = \frac{x+y}{x-y}

Solution:

Substitute v = \frac{y}{x} so y = vx and \frac{dy}{dx} = v + x\frac{dv}{dx} :

v + x\frac{dv}{dx} = \frac{1+v}{1-v}

Rearrange:

x\frac{dv}{dx} = \frac{1 + v}{1 - v} - v = \frac{(1 + v)^2}{1 - v}.

Separate variables:

\frac{1-v}{1+v^2} \, dv = \frac{dx}{x}

Integrate both the sides:

\int \frac{1-v}{1+v^2} \, dv = \int \frac{dx}{x}\Rightarrow \int \frac{1}{1+v^2} \, dv - \int \frac{v}{1+v^2} \, dv = \ln|x| + C.

Compute the integrals:

\tan^{-1}(v) - \frac{1}{2} \ln(1 + v^2) = \ln|x| + C.

Substitute v = y/x:

\tan^{-1}(\frac{y}{x}) - \frac{1}{2} \ln(1 + \frac{y^2}{x^2}) = \ln|x| + C.

Use In(1 + y²/x²) = In(x² + y²) - 2 In|x| terms cancel ):

\tan^{-1}(\frac{y}{x} )- \frac{1}{2} \ln(x^2 + y^2) = C'.

where C' is a constant.

\tan^{-1}(\frac{y}{x} )- \frac{1}{2} \ln(x^2 + y^2) = C.

Practise Questions

Questions 1. Solve the differential equation \frac{dy}{dx} = 3y using the separation of variables method.

Questions 2. Solve the first-order nonlinear differential equation y' = y^2 \sin(x).

Questions 3. Find the general solution of the first-order linear differential equation \frac{dy}{dx} + y = e^x .

Questions 4. Use the integrating factor method to solve \frac{dy}{dx} - \frac{y}{x} = x^2.

Questions 5. Solve the homogeneous differential equation \frac{dy}{dx} = \frac{x+y}{x-y}.

Questions 6. Solve the exact differential equation (x + y) \, dx + (x - y) \, dy = 0 .

Questions 7. Solve the exact differential equation (2xy - y²) dx + (x² - 2xy)dy = 0.

Questions 8. Solve the second-order differential equation y'' + 3y' + 2y = 0.

Questions 9. Determine the solution to the differential equation \frac{d^2y}{dx^2} + 4 \frac{dy}{dx} + 4y = 0 .

Questions 10. Solve the linear differential equation \frac{dy}{dx} + y \cot(x) = \csc(x).

Related Articles:

• Differential Equation
• Linear Differential Equation
• First Order Differential Equation
• Homogeneous Differential Equation
• Order and Degree Of Differential Equations
• Separation of Variables
• Exact Differential Equations