doraemonpaul

 40 Finding the Derivative of |x| using the Limit Definition 20 Particular solution to a Riccati equation $y' = 1 + 2y + xy^2$ 12 Solving 4th-order PDE 8 Integral $S_\ell(r) = \int_0^{\pi}\int_{\phi}^{\pi}\frac{(1+ r \cos \psi)^{\ell+1}}{(1+ r \cos \phi)^\ell} \rm d\psi \ \rm d\phi$ 8 Stochastic Heat Equation

### Reputation (15,045)

 +15 Solving $(x\cos y - y\sin y)dy - (x\sin y + y\cos y)dx =0$. +10 Transforming Differential Equation to a Kummer's Equation +10 Closed form solutions to linear 2nd order ODEs with polynomial coefficients. +10 I have a special solution for the Lane-Emden equation. Can I use it to find the general solution?

### Questions (36)

 8 About the great result of $\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx$ 8 $\mathcal{B}^{-1}_{s\to x}\{e^{as^2+bs}\}$ and $\mathcal{L}^{-1}_{s\to x}\{e^{as^2+bs}\}$ , where $a\neq0$ 8 How to find the general solution of $xy''-(2x+1)y'+x^2y=0$ when we know the general solution of $y''+2y'+xy=0$? 6 Why solving $\dfrac{\partial u}{\partial x}=\dfrac{\partial^2u}{\partial y^2}$ like this is wrong? 5 How to solve the nonlinear higher-order PDEs with overdetermined quantities of conditions whose their most-general solutions are known?

### Tags (179)

 443 ordinary-differential-equations × 537 49 limits × 5 387 partial-differential-equations × 459 45 functional-equations × 63 149 calculus × 107 44 definite-integrals × 24 94 integration × 65 43 laplace-transform × 53 50 derivatives × 13 33 special-functions × 31

### Bookmarks (3,030)

 51 Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$ 44 When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? 41 Computing the integral of $\log(\sin x)$ 28 Integrating $\int_0^ex^{1/x}\ \mathrm dx$ 28 Solve the functional equation $f(x)f(1/x)=f(x+1/x)+1$, $f(1)=2$, where $f(x)$ is a polynomial.