Solution to (D+2)(D+3)y=4t+5e^t with Initial Conditions y(0)=4, y'(0)=5

In summary, Daniel is trying to find the constants for C_3 + tC_4 + C_5e^t, but he is not able to find them.
  • #1
RadiationX
256
0
Solve the following differential equation:

[tex](D + 2)(D + 3)y = 4t + 5e^t; y(0)=4, y'(0)=5[/tex]

I have the following as the answer is it correct?

[tex]y=17e^{-2t} -13e^{-3t} + \frac{4}{11}t + \frac{5}{12}e^t[/tex]
 
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  • #2
You try plugging your answer back into the equation, to see if it works out right?
 
  • #3
Nope.Here's what Maple gives as the general solution

[tex]\frac{d^2 y}{dx^2}+5\frac{dy}{dx}+6y=4x+5e^x [/tex] Exact solution is :

[tex] y\left( x\right) =\frac{2}{3}x-\frac{5}{9}+\frac{5}{12}e^x+C_1 e^{-3x}+C_2 e^{-2x} [/tex]

I get some nasty looking coeff.

Daniel.
 
  • #4
i am way off then. by the way could MATLAB solve this problem?
 
  • #5
I'm not understanding what my professor calls an annihilator. what i have is

this D.E.

[tex](D + 2)(D + 3)y = 4t + 5e^t;y(0)=4,y'(0)=5[/tex]

I need to find a differential opperator that will make the right side of this D.E.

zero. So i think that this differential opperator (annihilator) should be this.

[tex]D^2(D-1)[/tex]

So to solve this problem i need to solve this homogenous D.E. first:

[tex](D + 2)(D + 3)y=0[/tex] and its solutions are this:

[tex]y_c=C_1e^{-2t} + C_2e^{-3t}[/tex]

now i need to find the other part which involves the differential opperator. so

now i need to solve this part [tex]D^2(D-1)=0[/tex]

which yields [tex] C_3 + tC_4 + C_5e^t[/tex]

if all of the above is correct i need to find out what the arbitrary constants of

[tex] C_3 + tC_4 + C_5e^t[/tex] are

so what i do is say that this [tex] C_3 + tC_4 + C_5e^t[/tex] is equal to

[tex]y_p[/tex]

so now i have [tex]y_p= C_3 + tC_4 + C_5e^t[/tex]

now i find [tex]y'_p[/tex] and [tex]y''_p[/tex]

[tex]y'_p= C_4 +C_5e^t [/tex]

[tex]y''_p=C_5e^t[/tex]

now what i do is plug in all the y-sub-p's into this [tex]y'' + 5y' +6y= 4t +5e^t[/tex]

and match up the the coef. to find out what the constants are

what i get is this

[tex]12C_5e^t + 5C_4 + 6tC_4 + 6C_3= 4t + 5e^t[/tex]

now my problem is that the C-sub-4's are not the same. one of them is

multiplied by t which is a problem because i can't get the coef. to match

what have i done wrong?
 
  • #6
You have this system

[tex] \left\{\begin{array}{c} 5C_{4}+6C_{3}=0\\6C_{4}=4 \end{array} \right [/tex]

Solve it.

Daniel.
 
  • #7
dextercioby said:
You have this system

[tex] \left\{\begin{array}{c} 5C_{4}+6C_{3}=0\\6C_{4}=4 \end{array} \right [/tex]

Solve it.

Daniel.


i don't see why could you elaborate?
 
  • #8
RadiationX said:
[tex]12C_5e^t + 5C_4 + 6tC_4 + 6C_3= 4t + 5e^t[/tex]

That should read

[tex] 12C_{5} e^{t}+6C_{4}t+\left(5C_{4}+6C_{3}\right)\equiv 5e^{t}+4t+0 [/tex]

Do you see where the system i posted comes from...?

Daniel.
 
  • #9
ahh! yes i see now. i made a simple mistake, but to be truthful i did not know that i could combine csub4 and csub3 but now i see why i can.
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It describes the relationship between a function and its rate of change at any given point.

2. What is the purpose of solving a differential equation problem?

The purpose of solving a differential equation problem is to find the function that satisfies the equation and describes the behavior of a system over time. This function can then be used to make predictions and analyze the system's behavior.

3. What are the different types of differential equations?

There are several types of differential equations, including ordinary differential equations, partial differential equations, and stochastic differential equations. Ordinary differential equations involve only one independent variable, while partial differential equations involve multiple independent variables. Stochastic differential equations involve random variables and are used to model systems with uncertainty.

4. What are some common techniques for solving differential equations?

The most common techniques for solving differential equations include separation of variables, integrating factors, and the method of undetermined coefficients. Other methods include Laplace transforms, power series, and numerical methods such as Euler's method or Runge-Kutta methods.

5. What are some real-world applications of differential equations?

Differential equations have many real-world applications, including modeling population growth, predicting weather patterns, analyzing the spread of diseases, and designing electrical circuits. They are also used in physics, engineering, economics, and many other fields to describe and understand complex systems and phenomena.

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