Intereting Posts

Derive quadratic formula
Parametric QR factorization: $\mathbf{D}(\alpha)\mathbf{V}^*$ a diagonal times a constant unitary matrix
Is the Jacobian a divergence of some vector field? (manifolds)
Prove that the boy cannot escape the teacher
LogSine Integral $\int_0^{\pi/3}\ln^n\big(2\sin\frac{\theta}{2}\big)\mathrm d\theta$
sum of series using mean value theorem
Proof of certain Gaussian integral form
writing down the minimal discriminant of an elliptic curve
Probability of getting A to K on single scan of shuffled deck
Subadditivity for Analytic Capacity Disjoint Compacts separated by a Line
Linear independency before and after Linear Transformation
Find all integers $x$, $y$, and $z$ such that $\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$
Limit $\lim_{x\to0^-}{(1+\tan(9x))^{\frac{1}{\arcsin(5x)}}}$
Sailors, monkey and coconuts
How to Show that Linear transformation is invertible?

Solve the partial differential equation $$u_{xx}+2u_{xy}-3u_{yy}=0$$ subjet to the initial conditions $u(x,0)=\sin{x}$, $u_{y}(x,0)=x$.

What I have done

$$

3\left(\frac{dx}{dy}\right)^2+2\frac{dx}{dy}-1=0

$$

implies

$$\frac{dx}{dy}=-1,\frac{dx}{dy}=\frac{1}{3}

$$

and so

$$

x+y=c_{1},\ 3x-y=c_{2}.

$$

Let $ \xi=x+y$, $\eta=3x-y$. Then

\begin{align}

u_{xx}&=u_{\xi\xi}+6u_{\xi\eta}+9u_{\eta\eta} \\ u_{yy}&=u_{\xi\xi}-2u_{\xi\eta}+u_{\eta\eta} \\

u_{xy}&=u_{\xi\xi}-2u_{\xi\eta}-3u_{\eta\eta}.

\end{align}

Applying substitutions,

$$

u_{\xi\eta}=0.

$$

Thus,

\begin{align}

u(\xi,\eta)&=\varphi(\xi)+\psi(\eta) \\

u(x,y)&=\varphi(x+y)+\psi(3x-y).

\end{align}

Applying the initial value condition,

\begin{align}

u(x,0)&=\varphi(x)+\psi(3x)=\sin{x} \\

u_{y}(x,0)&=\varphi'(x)-\psi'(3x)=x

\end{align}

Therefore,

\begin{align}

\varphi(x)&= \frac{1}{2} \left(\sin{x}+\int_{x_{0}}^{x} \tau \, d\tau \right)+\frac k2 \\

ψ(3x)&=\frac{1}{2} \left(\sin{x}-\int_{x_{0}}^{x}\tau \, d\tau \right)-\frac k2.

\end{align}

I have no idea how to get $ψ(x)$. Does anyone could help me to continue doing this question? Thanks very much!

- How to reduce higher order linear ODE to a system of first order ODE?
- The equation $(x-2xy-y^2)\frac{dy}{dx}+y^2=0$
- Given the Cauchy's problem: $y'' = 1, y(0) = 0, y'(0) = 0$. Why finite difference method doesn't agree with recurrence equation?
- Using variation of parameters, how can we assume that nether $y_1$, $y_2$ equal zero?
- Properties of sin(x) and cos(x) from definition as solution to differential equation y''=-y
- Practical applications of first order exact ODE?

- Regularity of a domain - definition
- The extension of smooth function
- With Euler's method for differential equations, is it possible to take the limit as $h \to 0$ and get an exact approximation?
- Technical question about Strichartz estimate's proof.
- Orthogonal Trajectories Using Polar Coordinates. Correct Calculations, Two Different Answers?
- Solution of nonlinear ODE: $x= yy'-(y')^2$
- Find the trajectories that follow drops of water on a given surface.
- Solving a semilinear partial differential equation
- Solving a second-order linear ODE: $\frac{d^2 y}{dx^2}+(x+1)\cdot \frac{dy}{dx}+5x^2\cdot y=0$
- If $f(x) + f'(x) + f''(x) \to A$ as $x \to \infty$, then show that $f(x) \to A$ as $x \to \infty$

After getting your general solution of

$$

u(x, y) = \phi(x + y) + \psi(3x – y)

$$

We can notice the following:

$$

\phi(x) + \psi(3x) = \sin x \\

\phi'(x) – \psi'(3x) = x

$$

Now we can differentiate our first equation to get

$$

\phi'(x) + 3\psi'(3x) = \cos x

$$

Do you see where to go from here? (try adding three of the second equation to the first equation)

I misunderstood what the issue was, leaving the above for the sake of the community. As for having $\psi(3x)$ and wanting $\psi(x)$ try putting $3x = y$ and then you’ll get an expression for $\psi(y)$ which you can then rewrite as $\psi(x)$.

- Same solution implying row equivalence?
- What is the intuitive way to understand Dot and Cross products of vectors?
- Direct way to show: $\operatorname{Spec}(A)$ is $T_1$ $\Rightarrow$ $\operatorname{Spec}(A)$ is Hausdorff
- Prove that a metric space is countably compact if and only if every infinite sequence in $X$ has a convergent subsequence.
- Coloring a Complete Graph in Three Colors, Proving that there is a Complete Subgraph
- How do I determine $\lim_{x\to\infty} \left$?
- Pushforward of Lie Bracket
- $C^{1}$ function such that $f(0) = 0$, $\int_{0}^{1}f'(x)^{2}\, dx \leq 1$ and $\int_{0}^{1}f(x)\, dx = 1$
- What is the symbol ''$\divideontimes$'' (DIVIDE TIMES) for?
- Difference of consecutive pairs of sequence terms tends to $0$
- “Descent” on binary quadratic forms?
- $W_n = \frac{1}{n}\sum\log(X_i) – \log(X_{(1)})$ with Delta method
- The set of points where two maps agree is closed?
- Number of Elements Not divisble by 3 or 5 or 7
- Why is cardinality of set of even numbers = set of whole numbers?