# Help solving the initial value problem

• middieman147
In summary: So, in order to find C, you would need to divide the constant by e^\sqrt{1}=e.In summary, the initial value problem 2(√x)y'+y+4(√x) ; y(1)=2 can be solved by first separating the x's and y's in order to integrate, then finding the integrating factor using p(x)=-1/(2√x) and q(x)=2. The solution is y=2e^{-√x}+C, which can be simplified to y=-4√x-4+C. To find the constant, C, the initial condition is used by setting x=1 and dividing the constant by e. The final
middieman147
Given:
Solve the initial value problem 2(√x)y'+y+4(√x) ; y(1)=2

I am having trouble separating the x's and y's in order to integrate. I keep coming up with:

dy/dx +y/(2(√x))=2...

What do I keep missing here? I am pretty sure you leave the y(1)=2 alone until you are finished with the integration, in which case you plug in x=1 and y=2 to solve for the constant. Is this the correct thought process for this problem?

Thanks.

Or can I substitute in the y(1) for the 2 on the right side of the equation and make all the other x's equal to 1?

let

and

find the integrating factor

and then use

hope that help :)

Ok so now I have:

y'-y/(2√(x))=2

p(x)=-1/(2√x) q(x)=2

u(x)=e∫(-1/(2√x))=e-√x

y=e√x(2e-√x)+C

y=2+C

this is the solution for all values of x with C=0, correct? Or am I forgetting a step?

Thanks

you firstly have to integrate the right side
before you multiply it by
!

Ok:

y=-4sqrt(x)-4+C with C=10 is the answer given the IC y(1)=2?

when u divide the right side after integrating by
, u should also divide the constant too so it should be

middieman147 said:
Given:
Solve the initial value problem 2(√x)y'+y+4(√x) ; y(1)=2
Do you mean $2\sqrt{x}y'+ y= 4\sqrt{x}$?

I am having trouble separating the x's and y's in order to integrate. I keep coming up with:

dy/dx +y/(2(√x))=2...

What do I keep missing here? I am pretty sure you leave the y(1)=2 alone until you are finished with the integration, in which case you plug in x=1 and y=2 to solve for the constant. Is this the correct thought process for this problem?

Thanks.

HallsofIvy said:
Do you mean $2\sqrt{x}y'+ y= 4\sqrt{x}$?

yes I do.

thanks.

Saeed.z said:
when u divide the right side after integrating by
, u should also divide the constant too so it should be

wouldn't that just be another constant?

so instead of writing Ce^sqrt(x), you could just consider it a new constant

middieman147 said:
wouldn't that just be another constant?

so instead of writing Ce^sqrt(x), you could just consider it a new constant
C is a constant, but $Ce^\sqrt{x}$ is definitely not a constant.

isnt that irrelevant since there is an IC? wouldn't the x in that expression =1 and then just equal C multiplied by another constant?edit: Are you saying i should divide 10 by e^1 to find the constant?

middieman147 said:
isnt that irrelevant since there is an IC? wouldn't the x in that expression =1 and then just equal C multiplied by another constant?

edit: Are you saying i should divide 10 by e^1 to find the constant?
No, that's not irrelevant.

In using the Initial Condition to evaluate the constant, C , you do set x=1, but in the overall solution there is a term $Ce^\sqrt{x}$, unless the Initial Condition gives C=0, which is not the case here.

## 1. What is an initial value problem?

An initial value problem is a mathematical equation that involves finding the value of a function at a specific starting point. The starting point is known as the initial value, and the goal is to find a solution to the equation that satisfies both the equation and the initial value.

## 2. How do I solve an initial value problem?

The most common approach to solving an initial value problem is to use a technique called separation of variables, where the equation is rearranged into a form where the variables can be separated and then integrated. Other methods include using undetermined coefficients or Laplace transforms.

## 3. What are the applications of solving initial value problems?

Solving initial value problems is essential in many fields of science and engineering, such as physics, chemistry, and economics. It is used to model and predict the behavior of systems and phenomena, such as population growth, radioactive decay, and heat transfer.

## 4. Can initial value problems have multiple solutions?

Yes, an initial value problem can have multiple solutions. This can happen when the equation has more than one unknown function or when the initial value is not unique. In these cases, additional conditions or constraints are needed to determine a unique solution.

## 5. What if I cannot find a closed-form solution to an initial value problem?

In some cases, an initial value problem may not have a closed-form solution, meaning it cannot be expressed in terms of elementary functions. In these situations, numerical methods can be used to approximate the solution, such as Euler's method or the Runge-Kutta method.

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