The initial value problem,$$y’=2\sqrt{y}\;\;\;\; , y(0)=a$$ has (1) A unique solution if $a<0$ (2) No solution if $a>0$ (3) Infinitely many solutions if $a=0$ (4) A unique solution if $a \ge 0$ This is what I have done so far- Here, $f(x,y)=2\sqrt{y}\;\;$ and consider the rectancular region $R=\{ (x,y)\in \Bbb R^2:|x|\le c ,|y-a | \le […]

Solve the initial value problem for the sequence $\left \{ u_{n}| n \in \mathbb{N} \right \}$ satisfying the recurrence relation: $u_n − 5u_{n-1} + 6u_{n−2} = 0 $ with $u_0 = 1$ and $u_1 = 1$. Ive gotten the general solution to be $u_n = A(2)^n + B(3)^n$. Once I sub the initial values: $u_0 […]

The domain we want to obtain the solution on is $x \in [0,1]$. Let’s write the second difference equation corresponding to this differential equation problem $\frac{y_{i-1}-2y_i+y_{i+1}}{h^2} = 1$ (is that step valid by the way?). The first condition is $ y_0 = 0 $. For the second condition we use the first forward difference: $\frac{y_1-y_0}{h} […]

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