## How many solutions does a second order differential equation have?

To construct the general solution for a second order equation we do need two independent solutions.

## What is the difference between first order and second order differential equations?

As for a first-order difference equation, we can find a solution of a second-order difference equation by successive calculation. The only difference is that for a second-order equation we need the values of x for two values of t, rather than one, to get the process started.

Why do second order differential equations have two solutions?

5 Answers. second order linear differential equation needs two linearly independent solutions so that it has a solution for any initial condition, say, y(0)=a,y′(0)=b for arbitrary a,b. from a mechanical point of view the position and the velocity can be prescribed independently.

Can a second order differential equation be linear?

General Form of a Linear Second-Order ODE that if p(t), q(t) and f(t) are continuous on some interval (a,b) containing t_0, then there exists a unique solution y(t) to the ode in the whole interval (a,b). linearly independent solutions to the homogeneous equation. homogeneous problem and any particular solution.

### Can second order differential equation have 3 solutions?

The answer to this question depends on the constants p and q. This is a quadratic equation, and there can be three types of answer: two real roots.

Why does a second order differential equation have two constants?

How many initial conditions are needed for a second order differential equation?

two initial conditions
The general solution to a second order ODE contains two constants, to be de- termined through two initial conditions which can be for example of the form y(x0) = y0,y (x0) = y0, e.g. y(1) = 2,y (1) = 6.

## What are the differential equation of first order?

A first order differential equation is an equation of the form F(t,y,˙y)=0. A solution of a first order differential equation is a function f(t) that makes F(t,f(t),f′(t))=0 for every value of t. Here, F is a function of three variables which we label t, y, and ˙y.

## How to solve a linear second order differential equation?

To solve a linear second order differential equation of the form . d 2 ydx 2 + p dydx + qy = 0. where p and q are constants, we must find the roots of the characteristic equation. r 2 + pr + q = 0. There are three cases, depending on the discriminant p 2 – 4q. When it is . positive we get two real roots, and the solution is. y = Ae r 1 x + Be r 2 x

Which is the general solution of the differential equation?

Generally, when we solve the characteristic equation with complex roots, we will get two solutions r 1 = v + wi and r 2 = v − wi. So the general solution of the differential equation is. y = e vx ( Ccos(wx) + iDsin(wx) )

What are the initial conditions of a second order equation?

Fact: The general solution of a second order equation contains two arbitrary constants / coefficients. To find a particular solution, therefore, requires two initial values. The initial conditions for a second order equation will appear in the form: y(t0) = y0, and y′(t0) = y′0.

### When to go straight from the differential equation?

When the discriminant p2 − 4q is positive we can go straight from the differential equation y = Ae (2 3x) + Be (−3 2x) x = − (−6) ± √ ( (−6)2 − 4×9× (−1)) 2×9 y = Ae (1 + √2 3)x + Be (1 − √2 3)x When the discriminant p2 − 4q is zero we get one real root (i.e. both real roots are equal).