Free online Second Order Differential Equation Calculator is designed to check the second order differential of the given expression and display the result within seconds. Provide your equation as the input value and hit the calculate button to get the second order derivatives along with work.

**Second Order Differential Equation Calculator: **Second order differential equation is an ordinary differential equation with the derivative function 2. Go to the below sections to know the step by step process to learn the Second Order Differential Equation with an example. The Handy Calculator tool provides you the result without delay.

Second Order Differential Equation is represented as d^2y/dx^2=f”’(x)=y’’. Have a look at the following steps and use them while solving the second order differential equation.

- Take any equation with second order differential equation
- Let us assume dy/dx as an variable r
- Substitute the variable r in the given equation
- It will form a binomial equation
- Solve the equation and find its factors
- Find the value of y

**Example:**

**Question: Solve d^2y/dx^2-10dy/dx+25y=0?**

Answer:

Given equation is

d^{2}y/dx^{2}-10dy/dx+25y=0

Let us take y=e^{rx} then

dy/dx=re^{rx}

d^{2}y/dx^{2}=r^{2}e^{rx}

Substitute these values in the equation

r^{2}e^{rx}-10re^{rx}+25e^{rx}=0

e^{rx} (r^{2}-10r+25)=0

r^{2}-10r+25=0

r^{2}-5r-5r+25=0

r(r-5)-5(r-5)=0

(r-5)(r-5)=0

r=5

So, we can say that y=e^{5x}

dy/dx=5e^{5x}

d^{2}y/dx^{2}=25e^{5x}

So,

d^{2}y/dx^{2}-10dy/dx+25y

=25e^{5x}-10*5e^{5x}+25e^{5x}

=25e^{5x}-50e^{5x}+25e^{5x}

=0

So, in this case our solution is

y=Ae^{5x}+Bxe^{5x}

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### FAQs on Second Order Differential Equation Calculator

**1. How many solutions does a second order differential equation have?**

To construct the general solution for a second order equation, we need to take two independent solutions. So, it has 2 solutions.

**2. How do you solve a second order differential equation?**

Consider any variable for the derivative and substitute that value in the equation. Find the roots of that variable and substitute the values in the given equation.

**3. How do you write a second order differential equation?**

The standard form of a second order differential equation is pd^{2}y/dx^{2}+qdx/dy+r=0. The mathematical operator can be either plus or minus.

**4. What is the difference between first and second order differential equations?**

An equation having only first derivatives is known as a first order differential equation and an equation containing a second derivative is called a second order differential equation. First order differential equation is represented as dy/dx while the second order differential equation representation is d^{2y}/dx^{2}.