# Second Order Differential Equation Calculator + Online Solver With Free Steps

The **Second Order Differential Equation Calculator** is used to find the initial value solution of second order linear differential equations.

The second order differential equation is in the form:

**L(x)y´´ + M(x)y´ + N(x) = H(x) **

Where **L(x)**, **M(x)** and **N(x)** are continuous functions of **x**.

If the function **H(x)** is equal to zero, the resulting equation is a **homogeneous** linear equation written as:

**L(x)y´´ + M(x)y´ + N(x) = 0 **

If **H(x)** is not equal to zero, the linear equation is a **non-homogeneous** differential equation.

Also in the equation,

\[ y´´ = \frac{ d^{ \ 2} \ y }{ d \ x^{2} } \]

\[ y´ = \frac{ d \ y }{ d \ x } \]

If **L(x)**, **M(x),** and **N(x)** are **constants** in the second order homogeneous differential equation, the equation can be written as:

**ly´´ + my´ + n = 0 **

Where **l**, **m**, and **n** are constants.

A typical **solution** for this equation can be written as:

\[ y = e^{rx} \]

The **first** derivative of this function is:

\[ y´ = re^{rx} \]

The** second** derivative of the function is:

\[ y´´ = r^{2} e^{rx} \]

Substituting the values of **y**, **y´**, and **y´´** in the homogeneous equation and simplifying, we get:

**$l r^{2}$ + m r + n = 0 **

Solving for the value of **r** by using the quadratic formula gives:

\[ r = \frac{ – \ m \pm \sqrt{ m^{2} \ – \ 4 \ l \ n } } { 2 \ l } \]

The value of ‘r’ gives **three** different **cases** for the solution of the second-order homogeneous differential equation.

If the discriminant $ m^{2}$ – 4 l n is **greater** than zero, the two roots will be **real** and **unequal**. For this case, the general solution for the differential equation is:

\[ y = c_{1} \ e^{ r_{1} \ x} + c_{2} \ e^{ r_{2} \ x} \]

If the discriminant is equal to **zero**, there will be **one real root**. For this case, the general solution is:

\[ y = c_{1} \ e^{ r x } + c_{2} \ x e^{ r x } \]

If the value of $ m^{2}$ – 4 l n is** less** than zero, the two roots will be **complex** numbers. The values of r1 and r2 will be:

\[ r_{1} = α + βί \ , \ r_{1} = α \ – \ βί \]

In this case, the general solution will be:

\[ y = e^{ αx } \ [ \ c_{1} \ cos( βx) + c_{2} \ sin( βx) \ ] \]

The initial value conditions **y(0)** and **y´(0)** specified by the user determine the values of c1 and c2 in the general solution.

## What Is a Second Order Differential Equation Calculator?

**The Second Order Differential Equation Calculator is an online tool that is used to calculate the initial value solution of a second order homogeneous or non-homogeneous linear differential equation.**

## How To Use the Second Order Differential Equation Calculator

The user can follow the steps given below to use the Second Order Differential Equation Calculator.

### Step 1

The user must first enter the second-order linear differential **equation** in the input window of the calculator. The equation is of the form:

**L(x)y´´ + M(x)y´ + N(x) = H(x) **

Here **L(x)**, **M(x),** and **N(x)** can be continuous **functions** or **constants** depending upon the user.

The function ‘H(x)’ can be equal to zero or a continuous function.

### Step 2

The user must now enter the **initial values** for the second-order differential equation. They should be entered in blocks labeled, **“y(0)”** and **“y´(0)”**.

Here **y(0)** is the value of **y** at **x=0**.

The value **y´(0)** comes from taking the **first derivative** of **y** and putting **x=0** in the first derivative function.

### Output

The calculator displays the output in the following windows.

#### Input

The input window of the calculator shows the input **differential equation** entered by the user. It also displays the initial value conditions **y(0)** and **y´(0)**.

#### Result

The Result’s window shows the **initial value solution** obtained from the general solution of the differential equation. The solution is a function of **x** in terms of **y**.

#### Autonomous Equation

The calculator displays the **autonomous form** of the second-order differential equation in this window. It is expressed by keeping the **y´´** on the left-hand side of the equation.

#### ODE Classification

ODE stands for **Ordinary Differential Equation**. The calculator displays the classification of differential equations entered by the user in this window.

#### Alternate Form

The calculator shows the** alternate form** of the input differential equation in this window.

#### Plots of the Solution

The calculator also displays the **solution plot** of the differential equation solution in this window.

## Solved Examples

The following example is solved through the Second Order Differential Equation Calculator.

### Example 1

Find the general solution for the second-order differential equation given below:

**y´´ + 4y´ = 0 **

Find the initial value solution with the initial conditions given:

** y(0) = 4 **

**y´(0) = 6 **

### Solution

The user must first enter the **coefficients** of the given second-order differential equation in the calculator’s input window. The coefficients of **y´´**, **y´**, and **y** are **1**, **4**, and **0** respectively.

The **equation** is homogeneous as the right-hand side of the equation is **0**.

After entering the equation, the user must now enter the **initial conditions** as given in the example.

The user must now “**Submit**” the input data and let the calculator compute the differential equation solution.

The **output** window first shows the input equation interpreted by the calculator. It is given as follows:

**y´´(x) + 4 y´(x) = 0 **

The calculator computes the differential equation **solution** and shows the Result as follows:

\[ y(x) = \frac{11}{2} \ – \ \frac{ 3 e^{- \ 4x} }{ 2 } \]

The calculator displays the **Autonomous Equation** as follows:

**y´´(x) = – 4y´(x) **

The ODE classification of the input equation is a second-order** linear** ordinary differential equation.

The **Alternate Form** given by the calculator is:

**y´´(x) = – 4y´(x) **

**y(0) = 4 **

**y´(0) = 6 **

The calculator also displays the** solution plot** as shown in figure 1.

*All the images are created using Geogebra.*