I don't understand. If the condition has an infinity of roots, then how are we supposed to plug those ## k ## values into the eigensolutions ##y_\lambda (x)## into ##C[y_\lambda]=1## in order to find the constant ## A ##?
Since the constant ## B=0 ##, we have ## y=Asin(\sqrt{\lambda}x) ## and using the boundary condition at ## x=1 ## gives ## (1-\lambda)y(1)+y'(1)=0\implies (1-\lambda)Asin(\sqrt{\lambda})+A\sqrt{\lambda}cos(\sqrt{\lambda})=0 ##. But then this means ## sin(\sqrt{\lambda})=0\implies...
So for part b), I've got ## y=Asin(\sqrt{\lambda}x)+Bcos(\sqrt{\lambda}x) ##, where ## A, B ## are constants. The condition ## y(0)=0 ## gives ## B=0 ## and the boundary condition at ## x=1 ## gives ## y(1)=0\implies 0=Asin(\sqrt{\lambda})\implies sin(\sqrt{\lambda})=0 ## since ## A\neq 0 ##...
a) Proof:
Let ## \lambda ## be the Lagrange multiplier.
Then the auxiliary functional is ## \overline{S}[y]=\alpha y(1)^2+\int_{0}^{1}\beta y'^2dx-\lambda (\gamma y(1)^2+\int_{0}^{1}w(x)y^2dx-1) ##.
This gives ## \overline{S}[y+\epsilon h]=\alpha (y(1)+\epsilon h(1))^2+\int_{0}^{1}\beta...
It really depends on the person/individual. Everyone is different. For me, I cannot study with music, mainly because I get distracted very easily, since my blood type is O.
Okay, so for part b) of this problem, I've got that ## v=\cosh(v)+B\sinh(v)\implies B\sinh(v)=v-\cosh(v)\implies B=\frac{v-\cosh(v)}{\sinh(v)} ##. After I substitute ## B=\frac{v-\cosh(v)}{\sinh(v)} ## into ## B^2-1=2\sinh(v)+2B\cosh(v) ##, I have ##...
a)
Consider the functional ## S[y]=\int_{0}^{v}(y'^2+y^2)dx, y(0)=1, y(v)=v, v>0 ##.
By definition, the Euler-Lagrange equation is ## \frac{d}{dx}(\frac{\partial F}{\partial y'})-\frac{\partial F}{\partial y}=0, y(a)=A, y(b)=B ## for the functional ## S[y]=\int_{a}^{b}F(x, y, y')dx, y(a)=A...
Can certainly bring advancement in OP's career, at least. Earning the third Bachelor's degree might be a challenge since many schools won't allow it, just as you mentioned earlier. After I graduated from high school, I started working and I self-study mathematics, as of now.
Yeah, instead of trying to obtain another Bachelor's degree in either mathematics or physics, I think the OP should earn either a Master's degree in criminology or psychology, since the OP already has Bachelor's degrees in those two majors. In addition, the Bachelor's degrees that the OP already...