Answer check: differential initial value problem

In summary, the conversation discusses solving an initial value problem with the equation y''-8y'+16y=0 and the initial conditions y(0)=2 and y'(0)=7. The solution involves finding the auxiliary equation, solving for r, and plugging in values to find the final equation y(x)=2e4x-xe4x. The conversation also mentions the importance of verifying that the solution satisfies the differential equation and initial conditions.
  • #1
bakin
58
0

Homework Statement


solve the IVP

y''-8y'+16y=0

y(0)=2
y'(0)=7


Homework Equations





The Attempt at a Solution



auxiliary equation:
r2-8r+16=0
(r-4)2=0
so we have r=4 with m=2

y(x)= c1e4x+c2xe4x

y(0)--> c1e0 + 0 = 2
c1=2
now we have:
y(x)=2e4x+c2xe4x

take first derivative
y'(x) = 8e4x+c2e4x+4c2xe4x (product rule)

y'(0) ----> 8e0+c2e0+ 0 = 7
8+c2=7
c2=-1

so our final equation is:

y(x)=2e4x-xe4x

Everything seems ok, but just wanted to run it by here to make sure it's all done correctly. thanks all!
 
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  • #2
Looks fine to me
 
  • #3
It's much easier to check that a potential solution actually works than it is to get the solution. For your problem all you need to do are the following:
  1. Verify that y(x) = 2e4x + xe4x satisfies y'' - 8y' + 16y = 0.
  2. Verify that y(0) = 2.
  3. Verify that y'(0) = 7.

If your solution satisfies the differential equation and initial conditions, you can bask in the warm glow of confidence that you nailed that problem.
 
  • #4
...I didn't even think of trying that. Thanks guys!
 

1. What is a differential initial value problem?

A differential initial value problem is a type of mathematical problem that involves finding a function that satisfies a given differential equation while also satisfying a set of initial conditions.

2. What is the difference between an ordinary and a partial differential initial value problem?

The main difference between an ordinary and a partial differential initial value problem is the number of independent variables. An ordinary differential initial value problem has only one independent variable, while a partial differential initial value problem has two or more independent variables.

3. How do you solve a differential initial value problem?

To solve a differential initial value problem, you must first find the general solution to the given differential equation. Then, you can use the initial conditions to determine the specific solution that satisfies both the differential equation and the initial conditions.

4. What are some real-life applications of differential initial value problems?

Differential initial value problems are commonly used in physics, engineering, and other scientific fields to model natural phenomena and predict outcomes. They can be used to study the motion of objects, heat transfer, electrical circuits, and many other systems.

5. Can a differential initial value problem have more than one solution?

Yes, a differential initial value problem can have multiple solutions. However, it is important to note that for a given set of initial conditions, there can only be one unique solution that satisfies both the differential equation and the initial conditions.

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