Find the general solution to the homogeneous differential equation d^2 y / d t^2 - 6 d y / d t =...
Question:
Find the general solution to the homogeneous differential equation {eq}\frac{d^2 y}{d t^2} - 6 \frac{d y}{d t } = 0.{/eq}
The solution can be written in the form {eq}y = C_1 e^{r_1 t} + C_2 e^{r_2 t} {/eq}
with {eq}r_1 < r_2 {/eq}
Using this form, {eq}r_1 = \ and \ r_2 = {/eq}
IHomogeneous Differential Equation:
The given second-order homogeneous differential equation can be solved by finding the roots of the auxiliary equation. Two linearly independent solutions are found out and their linear combination forms the complete solution to the given differential equation.
Answer and Explanation: 1
Become a Study.com member to unlock this answer! Create your account
View this answerGiven {eq}\displaystyle \frac{d^2 y}{d t^2} - 6 \frac{d y}{d t } = 0\\ y'' -6y'= 0, {/eq} the auxiliary equation becomes
{eq}m^2-6m=0\\ \Rightarro...
See full answer below.
Ask a question
Our experts can answer your tough homework and study questions.
Ask a question Ask a questionSearch Answers
Learn more about this topic:
Get access to this video and our entire Q&A library
Differential Calculus: Definition & Applications
from
Chapter 13 / Lesson 6
15K
This lesson explores differential calculus. It defines a differential and delves into the many uses of differential equations.
Related to this Question
- Find the general solution to the homogeneous differential equation d 2 y/ d t 2 - 6 d y d t + 9 y = 0 . The solution can be written in the form y = C 1 f 1 ( t ) + C 2 f 2 ( t )
- Find the general solution to the homogeneous differential equation ({d^2}y)/({dt}^2) - 1({dy)/({dt}) = 0. The solution can be written in the form y = C_1 e^{r_1 t} + C_2 e^{r_2 t} with r_1 < r_2. Usin
- Find the general solution to the homogeneous differential equation \frac{d^2y}{dt^2}+6\frac{dy}{dt}+9y=0 The solution can be written in the form
- Find the general solution to the homogeneous differential equation \frac{d^2y}{dt^2} - 9 \frac{dy}{dt} + 14y = 0 The solution can be written in the form y = C_1 e^{r_1t} + C_2 e^{r_2t} with r_1 < r_2
- Find the general solution to the homogeneous differential equation d^2y/dt^2 - 5dy/dt = 0. The solution has the form y = C_1f_1(t) + C_2f_2(t).
- Find the general solution to the homogeneous differential equation frac{d^2y}{dt^2} - 8 frac{dy}{dt} + 32y = 0 The solution has the form y = C_1 f_1(t) + C_2f_2(t) with f_1(t) = ? and f_2(t)
- Find the general solution to the homogeneous differential equation \frac{d^2 y}{dt^2} - 18 \frac{dy}{dt} + 130y = 0 The solution has the form y = C_1 f_1 (t) + C_2 f_2 (t) Left to your own devices
- 1. Find the general solution to the homogenous differential equation \dfrac{d^2y}{dt^2}+20\dfrac{dy}{dt}+100y=0 2. The solution can be written in the form y=C_1f_1(t)+C_2f_2(t) with f_1(t) = {Blank} a
- Find the general solution to the homogeneous differential equation d^2 y / dt^2 + 10 dy / dt + 25 y = 0 The solution can be written in the form y = C_1 f_1 (t) + C_2 f_2 (t). Find f_1 (t) and f_2 (t).
- Find the general solution to the homogeneous differential equation (d^2y)/(dx^2) +5 dy/dx + 6y = 0. The solution has the form y = C_1 f_1(x) + C_2 f_2(x). Find f_1(x) and f_2(x).
- Find the general solution to the homogeneous differential equation. \frac{d^2y}{dt^2} - 16 \frac{dy}{dt} + 68y = 0.
- Find the general solution of the homogeneous differential equation \frac{d^2y}{dt^2} + 10\frac{dy}{dt} + 25y = 0
- Find the general solution to the homogeneous differential equation: \frac{d^2y}{dx^2} - 22 \frac{dy}{dx} + 117y = 0
- Find the general solution to the following homogeneous differential equation: y'' + 2y' + 2y = 0
- Find the general solution to the homogeneous differential equation (d^2 y)/(dx^2) + 3(dy/dx) - 4y = 0.
- Find the general solution of the homogeneous differential equation y'' + 7y' + 10y = 0
- Find the general solution to the homogeneous differential equation d^2y dt^2 -5dy dt =0.
- Find the general solution of the homogeneous differential equation. \\ y'' - 7y' + 10 y = 24e^x
- Find the general solution of the homogeneous differential equation y'=x+y x-y .
- Find the general solution to the homogeneous differential equation. 4y'' + y' = 0.
- Find the general solution of the homogeneous differential equation y'' + 2y' + 5y = 0.
- Find the general solution to the homogeneous differential equation: (d^2 y)/(dt^2) - 8 dy/dt + 20y = 0.
- Find the general solution to the homogeneous differential equation \frac{d^2y}{dt^2} - 8\frac{dy}{dt} + 16y = 0
- Find the general solution to the homogeneous differential equation: 12y" - 11y' - 5y = 0.
- Find the general solution to the homogeneous differential equation \frac{d^2y}{dt^2} -0\frac{dy}{dt} + 0y= 0
- Find the general solution to the following homogeneous differential equation: y''' + y' = 0.
- Find the general solution to the homogeneous differential equation. y''' - y = 0.
- Find the general solution to homogeneous differential equation: (y^2 + y x) dx + x^2 dy = 0
- Find the general solution of the homogeneous differential equation y'' + 1y = 0
- Find the general solution to the homogeneous differential equation: d^2y dx^2 -9dy dx -10y=0.
- Find the general solution to the homogeneous differential equation d2y/dx2 - 5dy/dx + 4y = 0
- Find the general solution of the homogeneous differential equation y'' -4y' + 4y = 0
- Find the general solution to the homogeneous differential equation: y" - 8y' + 25y = 0.
- Find the general solution to the homogeneous differential equation \frac { d ^ { 2 } y } { d t ^ { 2 } } - 16 \frac { d y } { d t } + 164 y = 0.
- Find the general solution to the homogeneous differential equation: (d2y/dt2)+10(dy/dt)+24y=0
- Find the general solution of the non homogeneous differential equation y'' -4y' + 4y = 2e^{2x} + 4x -12
- Find the general solution for the following homogeneous differential equation. 3 x dy / dx - 3 y = 4 x sec (y / x)
- Find the general solution of the differential equation. Homogeneous: (x - y)y' = x + y
- Find the general solution to the homogeneous differential equation d^2 y/dt^2 - 20 dy/dt + 100y = 0. Write the solution in the form y = C_1 f_1(t) + C_2 f_2(t).
- Find the general solution of the homogeneous differential equation: x(x + y) dy/dx = y(x - y). Leave your answer in implicit form.
- Find the general solution of the homogeneous differential equation x(x + y)\frac{dy}{dt} = y (x - y). Leave the answer in implicit form.
- Find the general solution to the homogeneous differential equation { \frac{d^2y}{dt^2} - 16\frac{dy}{dt} + 145y = 0 } The solution has the form { y=C_1f_1(t) + C_2f_2(t) \\with, f_1(t) =...........
- Determine whether the given differential equation is homogeneous, and if so find the general solution. x(x + y)y' = y(x-y)
- Find the general solution to the homogeneous differential equation \frac{d^2y}{dt^2}-20\frac{dy}{dt}+125y=0. The solution has the
- Find the general solution to the homogenous differential equation y 8 y + 25 y = 0
- Find the general solution to the non-homogeneous differential equation y''+y'-6y=sinx
- Find the general solution to the non-homogeneous differential equation. 2 x^2 {d^2y} / {dx^2} - 10 x {dy} / {dx} - 14 y = 8 cos (3 x)
- Show that the differential equation dy/dx = (3y^2 + 10x^2)/(xy) is homogeneous. Then find the general solution.
- Find the general solution of the given non-homogenous differential equation: y'' - 5y' + 6y = 5e^{-2x}
- Find the general solution of the given non-homogenous differential equation: y'' + 4y = 1 + x + x^2
- Find a linear homogeneous differential equation with the general solution y=c1+c2e^2x+c3e^{-5}x
- Find the general solution to the homogeneous differential equation \frac{d^2y}{dt^2}-15\frac{dy}{dt}=0.
- Find the solution to the homogeneous differential equation 2 \frac{d^2y}{dx^2} + \frac{dy}{dx} - y = 0.
- a) Find a particular solution to the non homogeneous differential equation y'' + 4y' + 5y = -5x +5e^{-x} b) Find the most general solution to the associated homogeneous differential equation. c)
- Find the general solution to the homogeneous differential equation \dfrac{d^2y}{dt^2} + 2\dfrac{dy}{dt} - 3y = 0. The solution can be written in the form y = C_1e^{r_1t} + C_2e^{r_2t} with r_1 less than r_2. Using this form, r_1 = __________ and r_2 = __
- Find the general solution to the homogeneous differential equation: \\ \frac{d^2y}{dt^2}-6\frac{dy}{dt}+18y=0
- Find the general solution to the homogeneous differential equation \frac {d^2 y}{dt^2} - 12 \frac {dy}{dt} + 52y = 0 The solution has the form y = C_1 f_1 (t) + C_2 f_2(t)
- Find the general solution to the homogeneous second-order differential equation. 8y'' - 2y' - 3y = 0
- Find the general solution to the homogeneous second-order differential equation: y" - 8y' +25y = 0.
- Find the general solution to the homogeneous second-order differential equation. 12y'' - 11y' - 5y = 0
- Find the general solution to the homogeneous second-order differential equation. y'' - 4y' + 13y = 0
- Find the general solution of the differential equation by the method of homogenous substitution. (x^2-y^2)y' = 2xy
- Find a particular solution for the homogeneous differential equation given below. -y^2 dx + x(x + y) dy = 0, y(1) = 1.
- Find the general solution to the homogeneous differential equation {d^2y}{dt^2}-4 {dy}{dt}+4y=0.
- Find a particular solution to the non homogeneous differential equation y'' + 4y' + 5y = -10x + 3e^{-x}
- Find a particular solution to the non homogeneous differential equation y'' + 4y' + 5y = -5x + e^{-x}
- Consider the differential equation {x^2}y'' - xy' + y = x(8 + {1/ln(x)}). (A) Find the general solution of the related homogeneous differential equation. (B) Find the general solution of the non-homog
- Find the general solution to the homogeneous differential equation \frac{d^2y}{dt^2} + 8 \frac{dy}{dt} + 12y = 0 The solution can be written in the form y = C_1 e^{r_1 t} + C_2 e^{r_2 t} with r_1
- Show that the following differential equation is homogeneous and then find its solution x(x + y) dy/dx = y (x -y)
- Find the general solution to this homogeneous differential equation: dy dx = x+y x-y .
- Find the general solution to the homogeneous differential equation \frac{d^2 y}{dt^2} - 4 \frac{dy}{dt} + 29y = 0. The solution has the form y = C_1 f_1(t) + C_2 f_2(t). Find f_1(t), f_2(t). Write your answers so that the functions are normalized with the
- For the differential equation: \frac{d^2y}{dx^2} + 6 \frac{dy}{dx} + 10y = 5 \cos(3x) Write down the auxiliary equation and hence find the homogenous solution.
- a. Find a particular solution to the non-homogeneous differential equation y'' + 4y' + 5y = 10x + 5e^{-x}. b. Find the most general solution to the associated homogeneous differential equation. (Use
- Find the solution of the homogeneous part of the differential equation given by: y^4 - 2 y '''+ y'' = e^x + e^x \: cos(x) + 2x.
- Find the general solution of the differential equation. (Give your solution as an equation.) dr/ds = 7/2 s.
- Find the general solution to the differential equation. x_1 t is a solution to the differential equation t^2 x''+ 3tx' -3x = 0, x1 t = t
- For the following differential equation, the homogeneous solution is given by For the following differential equation y" - 4y' + 4y = e^3t the homogeneous solution is given by y_h(t) = C_1e^{2t} + C_2
- Find the solution of the differential equation y'' + 4y' + 3y = 0 if y 0 = 2, y' 0 = -1 and the general solution is given by y x = c1e^-x + c2e^-3x
- 1. Find a particular solution to the nonhomogeneous differential equation y" + 3y' - 4y = e^(5x) 2. Find the most general solution to the associated homogenous differential equation. Use c_1 and c
- Find the general solution of the homogeneous differential equation {y}' = \frac{x + y}{x - y}.|, u = \frac{y}{x}| then u| satisfies the DE x \frac{du}{dx} + u = | Separate variables to get du = | dx|
- (a) Find a particular solution to the nonhomogeneous differential equation y ?? + 4 y ? + 5 y = 10 x + 3 e ? x . (b) Find the most general solution to the associated homogeneous differential equatio
- Find the equation of the general solution of the differential equation y^6dy/dx + 2x^5y^7 = 3x^5. Your answer should be of the form y = f(x,C).
- Find the equation of the general solution of the differential equation (xe^y)\frac{dy}{dx} = 5(x^7e^{5x} + e^y) Your answer should be of the form y = f(x,C).
- Solve the differential equation below for the general solution. Give your answer in the form y = f (x). x dy / dx = 2y + 1 / x.
- (a) Find a particular solution to the nonhomogeneous differential equation y ?? + 4 y ? + 5 y = ? 5 x + 5 e ? x . y p = _____ (b) Find the most general solution to the associated homogeneous diffe
- Find the general solution of the given differential equation: y 2 y + y = e / ( 1 + 12 ) .
- A. Find the general solution of the given differential equation.
- Find the general solution of the given differential equation. x {d y/d x} + 3 y = x^3 - x
- Find the general solution of the given differential equation. (x^2 - 9)dy/dx + 6y = (x + 3)^2.
- Find the general solution of the given differential equation y" + 4y' = \csc2x.
- Find the general solution to the given differential equation. y''-2y'-3y=3t^2+4t-5+5cos(2t)+e^{(2t)}
- Find the general solution of the given differential equation. xdy/dx=3xe^x-y+6x^2
- Find a general solution to the given differential equation. 15y'' + 4y' - 3y = 0
- Find the general solution of the given differential equation. y''' - 4y'' - 16y' + 64y = 0 y(t) =
- Find the general solution of the given differential equation. y' = -y / t + cos (t^2)
- Find the general solution to the given differential equation. (D^2 + 3)(D + 1)^2 y = 0.
- Find the general solution of the given differential equation y' = -y/t + cos (t2)
- Find the general solution of the given differential equation: y' - (1/x)y = x + 1.
- Find the general solution of the given differential equation. y''' - 3y'' + 4y = 243x^2e^ 2x
- Find the general solution of the given differential equation. x^3y''' - 5x^2y'' + 9xy' = 32x^4
Explore our homework questions and answers library
Browse
by subject