Diff EQ- exponential raised to the exponential?

In summary, the conversation discusses solving the equation (x+xy^{2})dx + e^{x^{2}}y dy =0 through integration and substitution. The resulting solution is y = sqrt(e^{e^{-x^{2}}}) - C, with the caution to keep track of the constant values. It is also mentioned to check the solution by verifying that it satisfies the original differential equation.
  • #1
SmashtheVan
42
0

Homework Statement


Solve the equation:

[tex](x+xy^{2})dx + e^{x^{2}}y dy =0[/tex]

Homework Equations



n/a

The Attempt at a Solution



i) [tex]xdx(1+y^{2}) = -e^{x^{2}}y dy[/tex]
(multiply both sides by 2 to prepare for integration:

ii) [tex]\frac{-2xdx}{e^{x^{2}}}= \frac{2y dy}{1+y^{2}}[/tex]

integrate and get:
iii) [tex]\frac{1}{e^{x^{2}}}= ln(1+y^{2}) + C[/tex]

first integral was acheived from Maple, I am not sure if its what should be used here, but its what i have for now

iv) exponentiate both sides to remove natural log:

[tex]e^{e^{-x^{2}}}=1 + y^{2} + C[/tex]

v)combine constants, separate y

[tex] y^{2} = e^{e^{-x^{2}}} - C [/tex]

vi)square root of the system:

[tex]y=\sqrt{ e^{e^{-x^{2}}}} -C[/tex]Does this seem like a reasonable answer? I'm wary because of the occurance of the exponential function, raised to the exponential function.
 
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  • #2
One thing that's nice about differential equations is that when you get a solution, you can check it by seeing if it satisfies the DE.

As a side note, you show the same C in your last three equations. Actually these are all different constants, plus the third equation does not follow from the second. sqrt(a + b) != sqrt(a) + sqrt(b).
 
  • #3
Mark44 said:
As a side note, you show the same C in your last three equations. Actually these are all different constants, plus the third equation does not follow from the second. sqrt(a + b) != sqrt(a) + sqrt(b).

yea, I am still getting used to keeping track of my C's...thanks for that note

i don't know why i didnt think to substitute the solution back in. thanks!
 

1. What is a differential equation involving an exponential raised to the exponential?

A differential equation involving an exponential raised to the exponential is any equation where the dependent variable is related to its derivative by an exponential function with an exponential argument.

2. How do you solve a differential equation with an exponential raised to the exponential?

To solve a differential equation with an exponential raised to the exponential, you can use the method of separation of variables, or transform it into a linear equation by taking the logarithm of both sides.

3. What is the general form of a solution to a differential equation with an exponential raised to the exponential?

The general form of a solution to a differential equation with an exponential raised to the exponential is y = C*e^x, where C is a constant and x is the independent variable.

4. Can an exponential raised to the exponential be used to model real-world phenomena?

Yes, an exponential raised to the exponential can be used to model various real-world phenomena such as population growth, radioactive decay, and the spread of infectious diseases.

5. Are there any special techniques for solving differential equations with an exponential raised to the exponential?

Yes, there are special techniques such as the Laplace transform and the method of Frobenius that can be used to solve more complex differential equations involving an exponential raised to the exponential.

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