Solving a Differential Equation

In summary: I didn't realize that was a separate problemIn summary, the conversation discusses solving a separable differential equation using trigonometric reciprocal identities and finding a particular solution using arctan. The conversation also briefly touches on checking the solution using a calculator.
  • #1
gmmstr827
86
1

Homework Statement



Solve:
2 * √(x) * (dy/dx) = cos^2(y)
y(4) = π/4

Homework Equations



TRIGONOMETRIC RECIPROCAL IDENTITY: sec(u) = 1 / cos(u)
arctan( 1 ) = π / 4

The Attempt at a Solution



This is a separable differential equation.

2 * √(x) * (dy/dx) = cos^2(y)
2 * √(x) * dy = cos^2(y) * dx
[2 / cos^2(y)] * dy = [1 / √(x)] * dx
TRIGONOMETRIC RECIPROCAL IDENTITY: sec(u) = 1 / cos(u)
[2 * sec^2(y)] * dy = x^(-1/2) * dx

∫ [2 * sec^2(y)] * dy = ∫ x^(-1/2) * dx
2 * tan(y) = 2 * √(x) + C
y(x) = arctan( √(x) + C) <<< General Solution

NOTE: arctan( 1 ) = π / 4
y(4) = arctan( √(4) + C )
y(4) = arctan( 2 + C)
C = -1

y(x) = arctan( √(x) - 1 ) <<< Particular Solution

Is that all correct? Thank you!
 
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  • #3
Thanks! I just remembered that I can check these in my calculator as well... >.<
 
  • #4
derivative y=√x+√x
y'=?
 
  • #5
tinala said:
derivative y=√x+√x
y'=?

Um... what?

Well, if
y=√(x)+√(x)
then
y=2√(x)
and
y'=4x^(3/2)/3

but you need to separate the variables first, then integrate not derive, so I don't see how that's relevant...?
 
  • #6
gmmstr827 said:
Um... what?

Well, if
y=√(x)+√(x)
then
y=2√(x)
and
y'=4x^(3/2)/3

but you need to separate the variables first, then integrate not derive, so I don't see how that's relevant...?

under sqrtx is also +sqrtx
 
  • #7
tinala said:
under sqrtx is also +sqrtx

Don't hijack other problems with your own.
 
  • #8
Char. Limit said:
Don't hijack other problems with your own.

sorry
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It describes the rate of change of a variable in terms of the other variables in the equation.

2. Why are differential equations important?

Differential equations are important because they are used to model and solve real-world problems in various fields such as physics, engineering, economics, and biology. They provide a powerful tool for understanding and predicting the behavior of systems that change continuously.

3. How do you solve a differential equation?

There are several methods for solving differential equations, including separation of variables, integrating factors, and series solutions. The specific method used depends on the type of differential equation and its order. In general, the goal is to find a function that satisfies the equation and any given initial conditions.

4. What are initial conditions in a differential equation?

Initial conditions are values that are given for the dependent variable and its derivatives at a particular point. They are used to determine the specific solution to a differential equation, as there may be many possible solutions that satisfy the equation.

5. Can all differential equations be solved analytically?

No, not all differential equations can be solved analytically. Some equations do not have closed-form solutions and require numerical methods for approximation. Additionally, some equations may be too complex to solve using current mathematical techniques.

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