Verifying a solution to a DE (please check my work)

In summary, the student is trying to verify if y_1=1 and y_2=√t are solutions to the equation yy'' + (y')^2 = 0. They show their attempt at solving and realize their mistake in the exponent properties. They also mention a question about linear independence and the Wronskian.
  • #1
darryw
127
0

Homework Statement


Is y_1 = 1 and y_2 = root t solutions of the eqn: yy'' + (y')^2 = 0 ?

first solution works (i already verified)

2nd solution i get this:

(1/2)t^(-1) + (1/4)t^(1/4) which does not equal zero.

is this correct so far? the thing that confuses me is the question tells me to "verfiy that they are solutions" but one of them isnt. Either i mutiplied exponents incorrectly, or y_2 is not a solution.
thanks for any help.

Homework Equations





The Attempt at a Solution


 
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  • #2
Show us the 1st and 2nd derivatives of [itex] \sqrt {t} [/itex]
 
  • #3
y(t) = t^(1/2)

y'(t) = (1/2)t^(-1/2)

y''(t) = (1/4)t^(-3/2)

plug into equation..

t^(1/2)((1/4)t^(-3/2) + ( (1/2)t^(-1/2))^2

(1/2)t^(-1) + (1/4)t^(1/4)

thanks
 
  • #4
hang on sec.. i see something stupid
 
  • #5
it was property of exponents that was problem.. wow.
so y_2 = root t is also a solution

(1/4t) - (1/4t) = 0
 
  • #6
darryw said:
y(t) = t^(1/2)

y'(t) = (1/2)t^(-1/2)

y''(t) = (1/4)t^(-3/2)

You also have an issure withe the 2nd der. You lost a minus sign, it should read:
y"(t) = [itex] - \frac 1 4 t^{- \frac 3 2} [/itex]
Now it should all work out.

plug into equation..

t^(1/2)((1/4)t^(-3/2) + ( (1/2)t^(-1/2))^2

(1/2)t^(-1) + (1/4)t^(1/4)

thanks
 
  • #7
darryw said:
it was property of exponents that was problem.. wow.
so y_2 = root t is also a solution

(1/4t) - (1/4t) = 0

One other thing. (1/4t) is usually interpreted to mean (1/4) * t. If you want t in the denominator, write this as 1/(4t).
 
  • #8
next part asks about linear independence, but i already know this is just making sure wronskian doesn't equal zero. thanks for all the help
 

1. How do I verify that my solution to a differential equation is correct?

There are a few steps you can take to verify the correctness of your solution to a differential equation. First, you can substitute your solution into the original differential equation and see if it satisfies the equation. Additionally, you can check if your solution satisfies any initial conditions given in the problem. Another method is to graph your solution and see if it matches the expected behavior based on the differential equation and initial conditions.

2. What should I do if my solution does not satisfy the differential equation?

If your solution does not satisfy the differential equation, it is likely that there is an error in your work. Double check your calculations and make sure you have correctly applied any rules or techniques for solving differential equations. If you are still unable to find the error, it may be helpful to seek assistance from a tutor or professor.

3. Can I use technology to verify my solution to a differential equation?

Yes, technology can be a helpful tool in verifying the correctness of your solution to a differential equation. You can use graphing calculators or software to graph your solution and the original differential equation to compare them. Additionally, many software programs have built-in functions for solving differential equations, which can also be used to check your work.

4. Is it possible to have more than one solution to a differential equation?

Yes, it is possible to have more than one solution to a differential equation. This is known as the "general solution" and it includes all possible solutions to the equation. However, in most cases, a specific solution is required, which involves finding specific values for any arbitrary constants in the general solution based on any given initial conditions.

5. What should I do if I am still uncertain about my solution to a differential equation?

If you are still unsure about the correctness of your solution to a differential equation, it may be helpful to seek feedback from a peer or professor. They may be able to provide guidance or point out any errors in your work. It is also important to carefully read and understand the problem statement and any given initial conditions, as these can greatly impact the solution to the differential equation.

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