How to Find y'(0) from an Integral Equation

In summary: Without knowing more information about the functions involved, it is not possible to find a numerical value for y'(0). You can only express it in terms of y(0).
  • #1
PhMichael
134
0

Homework Statement



[tex]\int_{2}^{y(x)}e^{t^2+1}dt + \int_{x^2+3x}^{0}\frac{e^z}{1+z}=0[/tex]

I need so find [tex]y'(0)[/tex].


The Attempt at a Solution



[tex]\frac{d}{dx}\int_{2}^{y(x)}e^{t^2+1}dt =y'(x) \cdot e^{y(x)^2+1} [/tex]

[tex]\frac{d}{dx}\int_{x^2+3x}^{0}\frac{e^z}{1+z}=-\frac{e^{x^2+3x}}{x^2+3x+1}\cdot (2x+3)[/tex]

adding them and substituting x=0 yields:

[tex]y'(0) \cdot e^{y(0)^2+1} -3=0[/tex]

but how can i find y'(0) from them equation?
 
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  • #2
PhMichael said:

Homework Statement



[tex]\int_{2}^{y(x)}e^{t^2+1}dt + \int_{x^2+3x}^{0}\frac{e^z}{1+z}=0[/tex]

I need so find [tex]y'(0)[/tex].

The Attempt at a Solution



[tex]\frac{d}{dx}\int_{2}^{y(x)}e^{t^2+1}dt =y'(x) \cdot e^{y(x)^2+1} [/tex]

[tex]\frac{d}{dx}\int_{x^2+3x}^{0}\frac{e^z}{1+z}=-\frac{e^{x^2+3x}}{x^2+3x+1}\cdot (2x+3)[/tex]

adding them and substituting x=0 yields:

[tex]y'(0) \cdot e^{y(0)^2+1} -3=0[/tex]

but how can i find y'(0) from them equation?
[tex]y'(x)= \frac{d}{dx}\int_{2}^{y(x)}e^{t^2+1}dt + \frac{d}{dx}\int_{x^2+3x}^{0}\frac{e^z}{1+z}[/tex]

Go with solving...

[tex]y'(0) = y'(0) \cdot e^{y(0)^2+1} -3[/tex]

[tex]y'(0) =\frac{-3}{1-e^{y(0)^2+1}} [/tex]

That's it.
 
  • #3
but I need to find a number, not an expression involving y(0). how can y(0) be found?
 
  • #4
You don't know y(x), so you can't know y'(0).
 

1. What is the purpose of differentiating an integral?

Differentiating an integral allows us to determine the rate of change of a given function. It can also help us find the maximum and minimum values of a function, as well as determine the slope of a tangent line at a specific point on the function.

2. How is the process of differentiating an integral different from finding the derivative?

The process of differentiating an integral involves finding the derivative of the function within the integral, while finding the derivative of a function involves finding the general formula for the rate of change of that function.

3. Can we differentiate an indefinite integral?

Yes, we can differentiate an indefinite integral. Differentiating an indefinite integral will result in the original function, since the derivative of an integral is simply the function itself.

4. What is the relationship between differentiating an integral and finding the area under a curve?

Differentiating an integral and finding the area under a curve are inverse processes. When we differentiate an integral, we are essentially finding the slope of the function, while finding the area under a curve involves finding the total amount of space between the curve and the x-axis.

5. How can we use differentiation to solve real-world problems?

Differentiation allows us to analyze and understand how a certain quantity changes over time. By finding the derivative of a function, we can determine the rate of change of that quantity, which can be applied to various real-world scenarios such as calculating velocity, acceleration, and growth rates.

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