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Homogeneous linear second order differential equations
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Homogeneous linear second order differential equations
Lessons
A Linear Second Order Differential Equation is of the form:
a(x)dx2d2y+b(x)dxdy+c(x)y=d(x)
Or equivalently,
a(x)y′′+b(x)y′+c(x)y=d(x)
Where all of a(x),b(x),c(x),d(x) are functions of x.
A Linear Second Order Differential Equation is called homogeneous if d(x)=0. So,
a(x)y′′+b(x)y′+c(x)y=0
And a general constant coefficient linear homogeneous, second order differential equation looks like this:
Ay′′+By′+Cy=0
Let's suppose that both f(x) and g(x) are solutions to the above differential equations, then so is
y(x)=c1f(x)+c2g(x)
Where c1 and c2 are constants
Characteristic Equation
The general solution to the differential equation:
Ay′′+By′+Cy=0
Will be of the form: y(x)=erx
Taking the derivatives:
y′(x)=rerx
y′′(x)=r2erx
And inputting them into the above equation:
Ar2erx+Brerx+Cerx=0
So we will have: erx(Ar2+Br+C)=0
The equation Ar2+Br+C=0 is called the Characteristic Equation. And is used to solve these sorts of questions.
Solving the quadratic we will get some values for r:
Real Roots: r1≠r2
Complex Roots: r1,r2=λ±μi
Repeated Real Roots: r1=r2
Though let's deal with only real roots for now
As we will get two solutions to the characteristic equation:
y1(x)=er1x y2(x)=er2x
So all solutions will be of the form:
y(x)=c1erx+c2erx
a(x)dx2d2y+b(x)dxdy+c(x)y=d(x)
Or equivalently,
a(x)y′′+b(x)y′+c(x)y=d(x)
Where all of a(x),b(x),c(x),d(x) are functions of x.
A Linear Second Order Differential Equation is called homogeneous if d(x)=0. So,
a(x)y′′+b(x)y′+c(x)y=0
And a general constant coefficient linear homogeneous, second order differential equation looks like this:
Ay′′+By′+Cy=0
Let's suppose that both f(x) and g(x) are solutions to the above differential equations, then so is
y(x)=c1f(x)+c2g(x)
Where c1 and c2 are constants
Characteristic Equation
The general solution to the differential equation:
Ay′′+By′+Cy=0
Will be of the form: y(x)=erx
Taking the derivatives:
y′(x)=rerx
y′′(x)=r2erx
And inputting them into the above equation:
Ar2erx+Brerx+Cerx=0
So we will have: erx(Ar2+Br+C)=0
The equation Ar2+Br+C=0 is called the Characteristic Equation. And is used to solve these sorts of questions.
Solving the quadratic we will get some values for r:
Real Roots: r1≠r2
Complex Roots: r1,r2=λ±μi
Repeated Real Roots: r1=r2
Though let's deal with only real roots for now
As we will get two solutions to the characteristic equation:
y1(x)=er1x y2(x)=er2x
So all solutions will be of the form:
y(x)=c1erx+c2erx
- Introductiona)What are Homogeneous Linear Second Order Differential Equations? And what are some sorts of solutions to them?b)What is the Characteristic Equation?
- 1.Solving Homogeneous Linear Second Order Differential Equations
Find some general solutions to the following constant coefficient homogeneous linear second order differential equation:
2y′′+5y′−3y=0 - 2.Using the initial conditions find a particular solution to the following differential equation:
y′′+3y′−4y=0 y(0)=3,y′(0)=−2
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3.
Second Order Differential Equations
3.1
Homogeneous linear second order differential equations
3.2
Characteristic equation with real distinct roots
3.3
Characteristic equation with complex roots
3.4
Characteristic equation with repeated roots
3.5
Reduction of order
3.6
Wronskian
3.7
Method of undetermined coefficients
3.8
Applications of second order differential equations