Find The Particular Solution Of The Differential Equation Calculator

Find Particular Solution

This calculator finds the particular solution for a first-order linear differential equation of the form y' + ay = b, given an initial condition y(x₀) = y₀.

What is a Particular Solution of a Differential Equation Calculator?

A particular solution of a differential equation calculator is a tool designed to find a specific solution to a differential equation that satisfies given initial conditions or boundary conditions. Unlike a general solution, which includes arbitrary constants and represents a family of functions, a particular solution is a single function that fits the specific constraints provided. Our calculator focuses on first-order linear differential equations of the form y' + ay = b, using an initial condition y(x₀) = y₀ to determine the unique constant.

This type of calculator is useful for students, engineers, and scientists who need to solve differential equations arising in various fields like physics, engineering, biology, and economics, and require a specific function rather than a general family of solutions. It automates the process of applying initial conditions to the general solution.

Common misconceptions include thinking that a particular solution is the only solution or that it applies to all initial conditions. A particular solution is unique *only* for the given set of initial conditions.

Particular Solution of Differential Equation Formula and Mathematical Explanation (y' + ay = b)

We consider the first-order linear non-homogeneous differential equation:

y' + ay = b

where 'a' and 'b' are constants, and y' = dy/dx.

To find the general solution, we first find the integrating factor (IF), which is e^{∫a dx} = e^ax.

Multiplying the differential equation by the IF:

e^axy' + ae^axy = be^ax

The left side is the derivative of (y * e^ax) with respect to x:

d/dx (y * e^ax) = be^ax

Integrating both sides with respect to x:

y * e^ax = ∫be^ax dx = (b/a)e^ax + C

So, the general solution is:

y(x) = b/a + Ce^-ax

To find the particular solution, we use the initial condition y(x₀) = y₀:

y₀ = b/a + Ce^-ax₀

Solving for C:

C = (y₀ – b/a)e^ax₀

Substituting C back into the general solution:

y(x) = b/a + (y₀ – b/a)e^ax₀e^-ax = b/a + (y₀ – b/a)e^a(x₀-x)

This is the particular solution satisfying y(x₀) = y₀.

Variables Used

Variable	Meaning	Unit	Typical Range
a	Coefficient of y	Varies (e.g., 1/time)	-100 to 100
b	Constant term	Varies (e.g., units of y per time)	-100 to 100
x₀	Initial x-value	Units of x	-100 to 100
y₀	Initial y-value at x₀	Units of y	-100 to 100
C	Constant of integration	Units of y	Determined by initial conditions
x	Independent variable	Units of x	Any real number
y(x)	Dependent variable (solution)	Units of y	Any real number

Table of variables for the particular solution of y' + ay = b.

Practical Examples

Let's use the particular solution of differential equation calculator for real-world scenarios.

Example 1: Newton's Law of Cooling

The temperature T of an object cooling in an environment of temperature T_s can be modeled by T' + k(T – T_s) = 0, or T' + kT = kT_s. This is our form y' + ay = b with y=T, a=k, b=kT_s.

Suppose k = 0.1 (per minute), T_s = 20°C, and initially at t=0 (x₀=0), the object's temperature T(0) = 100°C (y₀=100). Find the temperature after 10 minutes (target x = 10).

• a = 0.1
• b = 0.1 * 20 = 2
• x₀ = 0
• y₀ = 100
• target x = 10

Using the formula y(x) = b/a + (y₀ – b/a)e^a(x₀-x):

T(10) = 2/0.1 + (100 – 2/0.1)e^0.1(0-10) = 20 + (100 – 20)e^-1 = 20 + 80 * e^-1 ≈ 20 + 80 * 0.3678 ≈ 20 + 29.43 = 49.43°C.

The calculator would show y(10) ≈ 49.43.

Example 2: RC Circuit

For an RC circuit with a constant voltage source V, the charge Q on the capacitor follows Q' + (1/RC)Q = V/R. This is y' + ay = b with y=Q, a=1/RC, b=V/R.

Let R=1000 Ω, C=0.001 F, V=10 V. Then a = 1/(1000*0.001) = 1, b = 10/1000 = 0.01. If initially Q(0)=0 (x₀=0, y₀=0), find the charge after 2 seconds (target x = 2).

• a = 1
• b = 0.01
• x₀ = 0
• y₀ = 0
• target x = 2

Q(2) = 0.01/1 + (0 – 0.01/1)e^1(0-2) = 0.01 – 0.01e^-2 ≈ 0.01 – 0.01 * 0.1353 ≈ 0.01 – 0.001353 = 0.008647 Coulombs.

Our particular solution of differential equation calculator would give y(2) ≈ 0.008647.

How to Use This Particular Solution of Differential Equation Calculator

1. Identify a, b, x₀, y₀: Your differential equation must be in the form y' + ay = b, and you need an initial condition y(x₀) = y₀.
2. Enter Coefficient 'a': Input the value of 'a'.
3. Enter Constant 'b': Input the value of 'b'.
4. Enter Initial x₀: Input the x-value from your initial condition.
5. Enter Initial y₀: Input the y-value from your initial condition y(x₀).
6. Enter Target x: Input the x-value where you want to find the solution y(x).
7. Calculate: The calculator updates automatically, or click "Calculate".
8. Review Results: The calculator will display the constant C, the particular solution formula, and the value of y at your target x. It also shows a plot of the solution.
9. Interpret: The primary result is y(target x), the value of your solution at the specified x. The graph shows the behavior of the solution.

Key Factors That Affect the Particular Solution

• Coefficient 'a': This determines the rate of change in the exponential term of the solution. A larger 'a' (if positive) means the transient part decays faster. It significantly affects the time constant in physical systems.
• Constant 'b': This influences the steady-state or equilibrium value (b/a) that the solution approaches as x goes to infinity (if a > 0).
• Initial Condition (x₀, y₀): These values anchor the solution curve, determining the specific value of the integration constant C and thus selecting one specific curve from the family of general solutions.
• The form of the equation: This calculator is specifically for y' + ay = b. Other forms (non-linear, higher-order) will have different solution methods and particular solutions.
• The value of x₀ – target x: The difference between the initial x and the target x determines how far along the solution curve you are evaluating, affecting the exponential term e^a(x₀-x).
• Sign of 'a': If 'a' is positive, the exponential term decays, and y approaches b/a. If 'a' is negative, it grows exponentially (unless the term multiplying it is zero).

Understanding these factors is crucial when using a particular solution of differential equation calculator and interpreting its results in a real-world context.

Frequently Asked Questions (FAQ)

What is a differential equation?

An equation that relates one or more functions and their derivatives. They describe how a quantity changes.

What's the difference between a general and a particular solution?

A general solution includes arbitrary constants (like C) and represents a family of functions that satisfy the differential equation. A particular solution is a single function from that family, obtained by using initial or boundary conditions to find specific values for the constants.

Can this calculator solve all differential equations?

No, this particular solution of differential equation calculator is specifically for first-order linear equations of the form y' + ay = b.

What if my 'a' is zero?

If a=0, the equation is y' = b, which integrates to y = bx + C. The calculator's formula involves b/a, so it won't work directly if a=0. In that case, y = y₀ + b(x – x₀).

What if I have boundary conditions instead of initial conditions?

This calculator uses an initial condition (value at one point). Boundary conditions (values at two different points) are typically used for second or higher-order equations and require different methods.

How accurate is the particular solution of differential equation calculator?

It provides an exact analytical solution for the given form y' + ay = b, subject to standard numerical precision of JavaScript.

What does the constant C represent?

C is the constant of integration that arises when solving the differential equation. Its value is determined by the initial conditions to give the particular solution.

Can I use this for second-order differential equations?

No, this calculator is only for the first-order linear form y' + ay = b. Second-order equations require different methods (e.g., characteristic equation).

Related Tools and Internal Resources

• General Solution Calculator: Find the general solution for some types of differential equations.
• Equation Solver: Solve various algebraic equations.
• Integration Calculator: Calculate definite and indefinite integrals.
• Differentiation Calculator: Find derivatives of functions.
• Understanding Differential Equations: A guide explaining the basics of differential equations.
• Graphing Calculator: Plot functions and visualize data.