Find the general solution of the given differential equations

$4y{}^{\u2033}-4{y}^{\prime}-3=0$

Pizzadililehz
2022-03-25
Answered

Find the general solution of the given differential equations

$4y{}^{\u2033}-4{y}^{\prime}-3=0$

You can still ask an expert for help

Cassius Villarreal

Answered 2022-03-26
Author has **11** answers

Given:

$4y{}^{\u2033}-4{y}^{\prime}-3=0$

Concept:

The non - homogenous ODE, second-order linear equation of type$ay{}^{\u2033}+b{y}^{\prime}$ =constant

$y={y}_{h}+{y}_{p}$ is the general solution of $ay{}^{\u2033}+b{y}^{\prime}$ =constant

The solution:

$y}_{h$ is the solution to the homogeneous ODE $ay{}^{\u2033}+b{y}^{\prime}=0$

$y}_{p$ the particular solution, is any function that satisfies the non-homogeneous equation

Solution:

$4y{}^{\u2033}-4{y}^{\prime}=0$

The characteristic equation:

$4{m}^{2}-4m=0$

$4m(m-1)=0$

$m=0$ and $m=1$

The solution when roots are$a=0$ and $b=0$ is $c}_{1}{e}^{ax}+{c}_{2}{e}^{bx$

$y}_{h}={c}_{1}{e}^{1x}+{c}_{2}{e}^{\left(0\right)x$

$y}_{h}={c}_{1}{e}^{x}+{c}_{2}{e}^{0$

$y}_{h}={c}_{1}{e}^{x}+{c}_{2$

Now,

${y}_{p}=-\frac{3}{4}t$

so,

$y={y}_{h}+{y}_{p}$

$y=\left({c}_{1}{e}^{x}\right)+{c}_{2})+(-\frac{3t}{4})$

$y={c}_{1}{e}^{x}+{c}_{2}-\frac{3t}{4}$

Answer:

$y={c}_{1}{e}^{x}+{c}_{2}-\frac{3t}{4}$

Concept:

The non - homogenous ODE, second-order linear equation of type

The solution:

Solution:

The characteristic equation:

The solution when roots are

Now,

so,

Answer:

Jeffrey Jordon

Answered 2022-03-31
Author has **2262** answers

asked 2021-11-13

Use implicit differentiation to find ∂z / ∂x and ∂z / ∂y.

${x}^{2}-{y}^{2}+{z}^{2}-2z=4$

asked 2021-12-17

asked 2021-12-21

A certain ellipse is centered at (0, 0) from which the major axis always equal to 16 and the minor axis is free to change as long as it obeys $a>b$ for ellipses. The directrices of this ellipse is perpendicular to the x - axis. Determine the differential equation which will satisfy the abovementioned set up.

asked 2020-12-16

Existence of Laplace Transform

Do the Laplace transforms for the following functions exist? Explain your answers. (You do not need to find the transforms , just show if they exist or not)

a)$f(t)={t}^{2}\mathrm{sin}(\omega t)$

b)$f(t)={e}^{{t}^{2}}\mathrm{sin}(\omega t)$

Do the Laplace transforms for the following functions exist? Explain your answers. (You do not need to find the transforms , just show if they exist or not)

a)

b)

asked 2021-12-27

Find the solution of the following Differential Equations $y-{y}^{\prime}-6y=0,y\left(0\right)=6,{y}^{\prime}\left(0\right)=13$ .

asked 2021-12-11

Using the Laplace Transform Table in the textbook and Laplace Transform Properties,find the (unilateral) Laplace Transforms of the following functions:

$t\mathrm{cos}\left({\omega}_{0}t\right)u\left(t\right)$

asked 2021-08-11

We give linear equations y=1.5x