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' = 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

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