themobius6s

2022-09-08

associated homogeneous linear differential equations

Can someone please explain how associated homogeneous linear differential equations work with an example?

Can someone please explain how associated homogeneous linear differential equations work with an example?

Mckenna Friedman

Beginner2022-09-09Added 10 answers

Step 1

Lets say you have the linear differential equation ${y}^{\u2033}+y=3x$;

The associated homogeneous equation is ${y}^{\u2033}+y=0$

The set of the solutions to the homogeneous equation is $\alpha \mathrm{cos}+\beta \mathrm{sin};\alpha ,\beta \in \mathbb{R}$

Step 2

One particular solution to the innitial equation is 3x.

Thus theset of solutions to the initial equation is $\alpha \mathrm{cos}x+\beta \mathrm{sin}x+3x;\alpha ,\beta \in \mathbb{R}$

Lets say you have the linear differential equation ${y}^{\u2033}+y=3x$;

The associated homogeneous equation is ${y}^{\u2033}+y=0$

The set of the solutions to the homogeneous equation is $\alpha \mathrm{cos}+\beta \mathrm{sin};\alpha ,\beta \in \mathbb{R}$

Step 2

One particular solution to the innitial equation is 3x.

Thus theset of solutions to the initial equation is $\alpha \mathrm{cos}x+\beta \mathrm{sin}x+3x;\alpha ,\beta \in \mathbb{R}$

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.