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# 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), n^{th}-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.

## Solved Examples on Differential Equations

**Example 1:** Find the solution for the first-order differential equation y' = x^{2 }and y(0) = 2 and verify it using the differential equation calculator.

**Solution**:

Given: y' = x^{2} and y(0) = 2

dy/dx = x^{2}

dy = x^{2} dx.

Integrate the given first order differential equation y(x) = x^{3} / 3 + C

y(0) = 2

y(0) = F(0) + C

2 = (0)^{3} / 3 + C

C = 2

y(x) = x^{3} / 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' = 3x
^{2}and y(0) = 5 - y' = secx and y(0) = 7

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