Solving Cauchy Prob: y'=sin(x+y+3) y(0)=-3

  • Thread starter Kalidor
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In summary, after trying to solve the equation y'=\sin (x+y+3) with initial condition y(0)=-3, the speaker substituted x+y+3 with u and used the formula \frac{d u}{dx} = 1+\sin(u) to find a general solution. After substituting x=0, they found that C=-2 and the particular solution is y(x) = -2\arctan(\frac{x}{x-2})-x-3.
  • #1
Kalidor
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[tex]y'=\sin (x+y+3)[/tex]
[tex] y(0)=-3 [/tex]

I tried substituting [tex] x+y+3=u [/tex] and solving I get
[tex] \tan (u(x)) - \sec (u(x)) = x [/tex]

but what the heck can I do now?
 
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  • #2
Kalidor said:
[tex]y'=\sin (x+y+3)[/tex]
[tex] y(0)=-3 [/tex]

I tried substituting [tex] x+y+3=u [/tex]

You get
[tex]\frac{d u}{dx} = 1+\sin(u)[/tex]

and then

[tex]\int\frac{d u}{1+\sin(u)} = x+C,[/tex]
where [tex]C[/tex] is an arbitrary constant, or

[tex]-\frac{2}{\tan[\frac{u}{2}]+1}= x+C[/tex]

so the general solution to your ODE is

[tex]y(x) = -2\arctan(\frac{2+x+C}{x+C})-x-3[/tex]

Substituting [tex]x=0[/tex] you find that [tex]C=-2[/tex], so particular solution with condition [tex]y(0)=-3[/tex]

[tex]y(x) = -2\arctan(\frac{x}{x-2})-x-3[/tex]
 
  • #3
Thanks, it seems fine now.
 

Related to Solving Cauchy Prob: y'=sin(x+y+3) y(0)=-3

1. What is Cauchy's method for solving differential equations?

Cauchy's method is a technique used to solve initial value problems for ordinary differential equations. It involves finding a unique solution to the differential equation by using the initial value as a starting point.

2. What is the first step in using Cauchy's method?

The first step in using Cauchy's method is to rewrite the differential equation in the form of dy/dx = f(x,y), where f(x,y) is a function of both x and y.

3. How do you solve for the general solution using Cauchy's method?

To find the general solution, the differential equation is integrated with respect to x, treating y as a constant. This will result in a solution in the form of y = φ(x,C), where C is the constant of integration and φ(x,C) is the general solution.

4. What is the role of the initial value in Cauchy's method?

The initial value is used to determine the value of the constant of integration, C. It is substituted into the general solution to find the particular solution that satisfies the initial condition.

5. How do you apply Cauchy's method to solve the given differential equation?

To solve the differential equation y' = sin(x+y+3), y(0)=-3, we first rewrite it in the form of dy/dx = f(x,y), which gives us dy/dx = sin(x+y+3). Then, we integrate both sides with respect to x to get the general solution y = -cos(x+y+3) + C. Finally, we substitute the initial value y(0)=-3 to find the particular solution y = -cos(x+y+3) - 3.

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