Find a Particular Solution to the Differential Equation Calculator
Differential Equation Calculator
Differential Equation Calculator calculates the solution for the given first-order differential equation when we know the initial condition. A differential equation is an equation that contains the derivative of a function.
What is Differential Equation Calculator?
Differential Equation Calculator is an online tool that helps to compute the solution for the first-order differential equation when the initial condition is given. A differential equation that has a degree equal to 1 is known as a first-order differential equation. To use this differential equation calculator , enter the values in the given input boxes.
Differential Equation Calculator
How to Use Differential Equation Calculator?
Please follow the steps below to find the solution of the first-order differential equation using the online differential equation calculator:
- Step 1: Go to Cuemath's online differential equation calculator.
- Step 2: Enter the values in the input boxes.
- Step 3: Click on the "Solve" button to find the solution.
- Step 4: Click on the "Reset" button to clear the fields and enter new values.
How Does Differential Equation Calculator Work?
A differential equation is defined as an equation that consists of the derivative of the dependent variable with respect to the independent variable. The rate of change of a quantity is represented by derivatives. Thus, a differential equation represents the relationship between a changing quantity and a change in another quantity. A differential equation can be classified into different types depending upon the degree. We can have first-order (degree = 1), second-order (degree = 2), nth-order (degree = n) differential equations. In a first-order differential equation, all the linear equations expressed in the form of derivatives are in the first order. Such an equation is given as y' = dy/dx = f(x, y). To find the solution of a first-order differential equation, when the initial condition y(0) is known, the steps are as follows:
- Express the given equation as dy/dx = f(x).
- Now write the equation as dy = f(x)dx.
- Integrate both sides of the function.
- We get the resultant as y = F(x) + C.
- To determine the value of C, substitute the values of the initial condition, y(0). Thus, y(0) = F(0) + C or C = y(0) - F(0).
- Now plug the value of C back into the equation given in step 4. This will be the solution to the differential equation.
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Solved Examples on Differential Equations
Example 1: Find the solution for the first-order differential equation y' = x2 and y(0) = 2 and verify it using the differential equation calculator.
Solution:
Given: y' = x2 and y(0) = 2
dy/dx = x2
dy = x2 dx.
Integrate the given first order differential equation y(x) = x3 / 3 + C
y(0) = 2
y(0) = F(0) + C
2 = (0)3 / 3 + C
C = 2
y(x) = x3 / 3 + 2
Example 2: Find the solution for the first-order differential equation y' = sinx and y(0) = 3 and verify it using the differential equation calculator.
Solution:
Given: y' = sinx and y(0) = 3
dy/dx = sinx
dy = sinx dx.
Integrate the given first order differential equation y(x) = -cosx + C
y(0) = 3
y(0) = F(0) + C
3 = -cos (0) + C
3 + 1 = C
C = 4.
y(x) = -cosx + 4
Now, try the differential equation calculator and find the solutions for:
- y' = 3x2 and y(0) = 5
- y' = secx and y(0) = 7
☛ Math Calculators:
Find a Particular Solution to the Differential Equation Calculator
Source: https://www.cuemath.com/calculators/differential-equation-calculator/