Solving differential equations

In summary, the conversation is about solving a differential equation using a formula for tan(a+b). The answer is found by taking the tangent of both sides and using the formula to simplify. The final solution is y=(x+1)/(1-x).
  • #1
physstudent1
270
1
[SOLVED] Solving differential equations

Homework Statement



Find the solution:
dy/dx = (1+y^2)/(1+x^2)
hint: there is a formula tan(a + b) that will simplify your answer.
y(0)=1

Homework Equations





The Attempt at a Solution



I separated the variables and integrated ending up with

arctan(y)=arctan(x)+c
I do not see how to take this to the answer below. I understand how to use the formula they gave me in the hint but I do not see how it applies here...

the answer is
y=(1+x)/(1-x) this problem is from my exam review we have the solutions but not how to get them...
 
Physics news on Phys.org
  • #2
tan(arctan(y))=y

take the tangent of both sides and then the formula for tan(a+b) goes to work since you have tan(arctan(x)+c), which is also (tan(a)+tan(b))/(1-tan(a)tan(b)) so you get (x+tan(c))/(1-x*tan(c)).

y(0)=1 --> tan(c)=1 --> c= arctan(1)

and you get

y=(x+tan(c))/(1-x*tan(c)) --> (x+1)/(1-x)
 
  • #3
thanks a lot I was thinking you had to take the tan of both sides but I wasn't sure you cleared it up nicely.
 

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives to represent the rate of change of a variable with respect to another variable.

2. What is the purpose of solving differential equations?

The purpose of solving differential equations is to find the function or functions that satisfy the equation. This can help in understanding and predicting the behavior of systems in various fields such as physics, engineering, and economics.

3. What methods are used to solve differential equations?

There are various methods used to solve differential equations, including separation of variables, substitution, and using power series. Other methods include Laplace transforms, numerical methods, and using software programs such as MATLAB.

4. Can all differential equations be solved analytically?

No, not all differential equations can be solved analytically. In some cases, it may not be possible to find the exact solution, and numerical methods may be used instead. Additionally, some differential equations may have no solutions or an infinite number of solutions.

5. What are some real-world applications of solving differential equations?

Differential equations have numerous real-world applications, including modeling population growth, predicting weather patterns, and analyzing electrical circuits. They are also used in fields such as economics, biology, and chemistry to understand complex systems and make predictions.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
388
Replies
7
Views
439
  • Calculus and Beyond Homework Help
Replies
8
Views
693
  • Calculus and Beyond Homework Help
Replies
2
Views
801
  • Calculus and Beyond Homework Help
Replies
1
Views
771
  • Calculus and Beyond Homework Help
Replies
4
Views
874
  • Calculus and Beyond Homework Help
Replies
7
Views
624
  • Calculus and Beyond Homework Help
Replies
21
Views
736
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
973
Back
Top