Extension: Solving y'' + ay' + by = 0
Read this topicExplanations 3
- This is a 2nd order linear homogeneous differential equation with constant coefficients. Let's look at those properties one at a time. Each term with yyy or its derivatives is o…
By Calvin Lee, Ahaan Rungta
- 0=y′′+ay′+by0 = y'' + ay' + by0=y′′+ay′+by To solve this equation, we look at the characteristic polynomial, which for this equation is r2+ar+b=0.r^2 + ar + b = 0.r2+ar+b=0. The…
By Calvin Lee
- We now know the solution for this second-order differential equation, which depends on the solutions to the characteristic polynomial ar2+br+c=0ar^2+br+c=0ar2+br+c=0. These are …
By Ritesh Ragavender, Alex Federspiel
Problems 6
- Solve the differential equation y′′+6y′+9y=0y'' + 6y' + 9y = 0y′′+6y′+9y=0 where y(0)=8y(0) = 8y(0)=8 and y′(0)=−1y'(0) = -1y′(0)=−1.
- Solve the differential equation y′′+7y′+12y=0y'' + 7y' + 12y = 0y′′+7y′+12y=0 where y(0)=−7y(0) = -7y(0)=−7 and y′(0)=2y'(0) = 2y′(0)=2.
- There is a unique twice-differentiable function ggg on the positive reals such that g(x)+2x3g′(x)+x4g′′(x)=0g(x) + 2x^3 g'(x) + x^4 g''(x) = 0g(x)+2x3g′(x)+x4g′′(x)=0 for all po…
- Which of the following twice-differentiable real functions f :R→Rf \colon \mathbb{R} \to \mathbb{R}f:R→R satisfy the "difference of squares" functional equation: f(x)2−f(y)2=f(…
- Solve the differential equation y′′+y′+y=0y'' + y' + y = 0y′′+y′+y=0 where y(0)=2y(0) = 2y(0)=2 and y′(0)=1y'(0) = 1y′(0)=1.
- You've probably heard of simple harmonic motion. When a spring is damped, which typically means a frictional force is present, we can write a differential equation for its motio…