# Search results

1. ### Solving second order system

Your second-order equation y^{\prime \prime} - \frac{y^{\prime 2}}{y} + \frac{y^3}{C^2} = 0 probably has many solutions, but there is a simple solution of the form y(\lambda) = \pm i \alpha C \sec{(\alpha \lambda + \beta)} where \alpha and \beta are constants. You can show this very...
2. ### First Order ODE

This is a simple Bernoulli equation. You may want to google it's solution, or refer to any Calc textbook.
3. ### 1st order nonlinear ODE

Is this a research question, or just homework? If it's serious, I may be able to get somewhere with an analytic (explicit) solution... possibly.
4. ### Tips on good N-L-ODE

Sorry, I misunderstood -- the equation I quoted has no known general solution (analytical), but I suspect there are many numerical methods already associated with it. If you apply your method, you should then do a survey on the web of other numerical techniques applied this class of equations...
5. ### Tips on good N-L-ODE

How about this Riccati equation: y^{\prime} + y^2 + \alpha(x) = 0 (where alpha is an arbitrary function of x, and y = y(x) as well). This has no general solution (as far as I know) -- and it is very important. If you can provide an analytic solution to this, then fame and fortune is yours. ;-)
6. ### Almost Kummer's Equation

Take your equation, and make the change of variable \tau = 2 x This means that y^{\prime}_{x} = 2 y^{\prime}_{\tau} and y^{\prime \prime}_{xx} = 4 y^{\prime \prime}_{\tau \tau} Substitute these into your equation, and it becomes \tau y^{\prime \prime}_{\tau \tau} + (b -...
7. ### Coupled Differential Equations

Use your first equation to isolate y, namely, y = \sin{\omega t} - x^{\prime} - x Now, differentiate this to get y prime, y^{\prime} = \omega \cos{\omega t} - x^{\prime \prime} - x^{\prime} and substitute these into your second equation to get... \omega \cos{\omega t} - x^{\prime...
8. ### Solution to 1st order nonlinear differential equation

I see what you're saying -- and you can also repeat this process and provide a sort of superposition of these solutions, for example: For any solution, v_0, of the Ricatti equation v^{\prime } + v^2 + \Psi = 0 we can show, through differentiation, that there will always be another...
9. ### Reduce order of ODE y =c-2y'2/y

Any second-order equation of the form y^{\prime \prime} + \alpha(y) y^{\prime 2} + \beta(y) = 0 (where the derivative is with respect to 'x') may be converted into a first order equation of the form \frac{du}{dy} + 2 \alpha(y) u + 2 \beta(y) = 0 with the simple substitution u =...
10. ### Solving diff. eq.

You're missing two partial symbols. Are they supposed to be: \lambda f(y)= i m y \frac{\partial f(y)}{\partial y} + \frac{\partial g(y)}{\partial y} -\frac{k}{y}g \lambda g(y)= i m y \frac{\partial g(y)}{\partial y} - \frac{\partial f(y)}{\partial y} -\frac{k}{y}f ??? Also, if f and g only...
11. ### Solve y'y'''=y''

Sorry that should read: y(x) = \frac{1}{\alpha}\int{exp(Ei^{-1}(\alpha x + \beta)) dx}
12. ### Solve y'y'''=y''

Re-arrange your equation as y^{\prime \prime \prime} = \frac{y^{\prime \prime}}{y^{\prime}} Now integrate with respect to x to get y^{\prime \prime} = \kappa + \ln{y^{\prime}} where \kappa is a constant of integration. Now re-arrange and integrate to get \int{\frac{d...
13. ### Solution to the nonlinear 2nd order d.e

You can solve the cubic v equation with Vieta's Substitution. {Wolfram.com link}
14. ### Better substitution?

There is a solution that does not involve a substitution... if that's any help... First, multiply through by x + y^2, to get x y^{\prime} + y^2 y^{\prime} = y rearrange to get x y^{\prime} - y = -y^2 y^{\prime} but x y^{\prime} - y = y^2 ( \phi - \frac{x}{y})^{\prime} (where \phi is a...
15. ### A DE, maybe easy

Rearrange to get \frac{f^{\prime}}{f} = - \frac{x}{x^2 - a^2} now integrate to get \ln{f} = -\frac{1}{2} \ln{(x^2 - a^2)} + C
16. ### Separable Equations

Multiply both sides by \sec^2{y}, to get y^{\prime} \sec^2{y} = x Now, integrate both sides w.r.t. x to get \tan{y} = \frac{x^2}{2} + \kappa where \kappa is a constant.
17. ### Name this function!

Don't know what it's called, but I gave a solution of it elsewhere[sosmath.com] (sorry 'bout the plug for a rival website :wink: ).
18. ### Differential equation system, got stuck in a physics problem

No worries, mate. :cool:
19. ### Differential equation system, got stuck in a physics problem

I've only solved it for \kappa =0. For \kappa \ne 0, it is still a nasty mess. And, no, I'm not a student.
20. ### Differential equation system, got stuck in a physics problem

So, as r^{\prime}(t) = \frac{1}{2} \frac{(2 x(t) x^{\prime}(t) + 2 y(t) y^{\prime}(t))}{\sqrt{x^2(t) + y^2(t)}} = \frac{x(t)x^{\prime}(t) + y(t) y^{\prime}(t)}{\sqrt{x^2(t) + y^2(t)}} You can write m z^{\prime \prime}(t) = -q r^{\prime}(t) Hence, for z you get z^{\prime}(t) =...
21. ### How many solutions?

Hm. All I can think of is that your equation may be derived from y y^{\prime \prime} + A x + B - \frac{y^{\prime 2}}{2} = 0 where B is a constant. But y y^{\prime \prime} - \frac{y^{\prime 2}}{2} = 2 y^{\frac{3}{2}} (\sqrt{y})^{\prime \prime} Therefore you can re-write your equation...
22. ### How are order and degree defined for this DE? (cos(y'') + xy' = 0)

Yes, but it is effectively a first order equation, as all you have to do is let y^{\prime} = u(x).
23. ### Stuck with an integral

No, you're right -- I should have made this clear at the start. Sorry folks.
24. ### Stuck with an integral

Fair enough, but it wasn't a guess -- I got there by trying to find out what function, f(x), when divided by (x \sin{x} + \cos{x}) yields the integrand in part of its derivative. (The answer of course is f(x) = -x \sec{x}). Then it's a question of seeing if you can integrate the other part(s) of...
25. ### Stuck with an integral

To solve your integral, you can start by differentiating -\frac{x \sec{x}}{(x \sin{x} + \cos{x})} this will give you -(\frac{x \sec{x} }{x \sin{x} + \cos{x}})^{\prime} = \frac{x^2}{(x \sin{x} + \cos{x})^2} - \frac{(\sec{x} + x \sec{x} \tan{x})}{(x \sin{x} + \cos{x})} Now, you'll recognize...
26. ### Solution of First Order DE using Integrating Factors

I'm not sure that integrating factors will be of any use here. But you can still solve it, i.e. by making the substitution y = x + \sqrt{v} this will give you 1 + \frac{1}{2}\frac{v^{\prime}}{\sqrt{v}} = \frac{1 - x^2 - x \sqrt{v}}{-x \sqrt{v}} = (x - \frac{1}{x}) \frac{1}{\sqrt{v}} + 1...
27. ### Non linear differential equation

Is it not a^{\prime \prime} + 2 a^{\prime 2} + \frac{2a}{r} = L e^{-2a} and, presumably the derivative is with respect to r, yes?
28. ### DE from Theory of Vibrations

Yup. Have a look here.
29. ### Solve x'' = cox(x)

You cannot neglect the constant of integration. if y^{\prime \prime} = \cos{y} multiply by y^{\prime} gives you y^{\prime} y^{\prime \prime} = y^{\prime} \cos{y} integrate, and you get \frac{1}{2}y^{\prime 2} = \sin{y} + C Note the constant of integration which must not...
30. ### Second order homogenous with variable coeffecients

Err... if you want ot find out what kind of second-order equations are soluble, you can look here. They also have some solutions of PDEs on other pages.