Solution of Initial Value Problem

In summary, to determine the solution of the IVP y' + 4ty = 4t, y(0) = 6, the first step is to identify p(t) and g(t) as 4t and 4t respectively. Then, the integrating factor μ(t) is calculated by taking the integral of p(t), which results in e^(2t^2). However, when using the integrating factor method, it is important to remember to multiply both sides of the equation by e^(2t^2) * g(t), not just g(t). This mistake was corrected by Tim-Tim, who is a super hero.
  • #1
DrunkApple
111
0

Homework Statement


Determine the solution of the IVP y' + 4ty = 4t, y(0) = 6


Homework Equations





The Attempt at a Solution


p(t) = 4t
g(t) = 4t

μ(t) = e[itex]^{\int4tdt}[/itex]
= e[itex]^{\int p(t)}[/itex]
= e[itex]^{\int4tdt}[/itex]
= e[itex]^{2t^{2}}[/itex]

is this all I need? because i did
[itex]\frac{d}{dt}[/itex](y * μ(t)) = p(t) * g(t)
and the professor made a big red X mark on it, so I am confused.
 
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  • #2
Hi DrunkApple! :smile:
DrunkApple said:
p(t) = 4t
g(t) = 4t
…
[itex]\frac{d}{dt}[/itex](y * μ(t)) = p(t) * g(t)

nooo :redface: …

you're multiplying the whole equation by e2t2,

so the RHS should be e2t2 * g(t), shouldn't it? :wink:​
 
  • #3
wow...
can't believe i made that mistake...
thank you super hero tim-tim
 

FAQ: Solution of Initial Value Problem

What is an initial value problem?

An initial value problem is a mathematical problem that involves finding the solution to a differential equation, given an initial condition. It typically involves finding the function that satisfies the equation and the initial condition at a specific point or set of points.

How do you solve an initial value problem?

There are various methods for solving initial value problems, including analytical techniques such as separation of variables or variation of parameters, as well as numerical methods like Euler's method or the Runge-Kutta method. The chosen method depends on the complexity of the problem and the desired level of accuracy.

What is the importance of initial value problems in science?

Initial value problems are essential in science as they help us model and understand natural phenomena. Many physical processes, such as radioactive decay or population growth, can be described using initial value problems. They also play a crucial role in fields such as engineering, physics, and economics.

Can initial value problems have multiple solutions?

No, initial value problems generally have a unique solution. This is because the initial condition acts as a constraint that narrows down the possible solutions to only one. However, there can be special cases where multiple solutions exist, known as singular solutions.

How can I check if my solution to an initial value problem is correct?

To check the accuracy of a solution to an initial value problem, you can plug it back into the original differential equation and see if it satisfies the equation at all points. You can also compare your solution to a known solution or use numerical methods to verify the results.

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