Question

In: Advanced Math

Consider the ODE y''(x) = λy(x) for some real constant λ. Determine ALL values of λ...

Consider the ODE y''(x) = λy(x) for some real constant λ. Determine ALL values of λ for which there exists solutions satisfying the boundary conditions y(0) = y(10) = 0. For each such λ, give all possible solutions. Are they unique?

Solutions

Expert Solution


Related Solutions

y''−2xy' + λy = 0, −∞ < x < ∞ where λ is a constant, is...
y''−2xy' + λy = 0, −∞ < x < ∞ where λ is a constant, is known as the Hermite equation, named after the famous mathematician Charles Hermite. This equation is an important equation in mathematical physics. • Find the first four terms in each of two solutions about x = 0 and show that they form a fundamental set of solutions • Observe that if λ is a nonnegative even integer, then one or the other of the series...
For which values of λ does the system of equations (λ − 2)x + y =...
For which values of λ does the system of equations (λ − 2)x + y = 0 x + (λ − 2)y = 0 have nontrivial solutions? (That is, solutions other than x = y = 0.) For each such λ find a nontrivial solution.
Determine the eigenvalues and the corresponding normalized eigenfunctions of the following Sturm–Liouville problem: y''(x) + λy(x)...
Determine the eigenvalues and the corresponding normalized eigenfunctions of the following Sturm–Liouville problem: y''(x) + λy(x) = 0, x ∈ [0;L], y(0) = 0, y(L) = 0,
x^2 y ′′ + xy′ + λy = 0 with y(1) = y(2) and y ′...
x^2 y ′′ + xy′ + λy = 0 with y(1) = y(2) and y ′ (1) = y ′ (2) Please find the eigenvalues and eigenfunctions and eigenfunction expansion of f(x) = 6.
Let the following ODE be given. Let x be a real number. xy"+(1-x)y' = (k-1/2)y Give...
Let the following ODE be given. Let x be a real number. xy"+(1-x)y' = (k-1/2)y Give an explict solution for k=-2.5
Consider the following first-order ODE dy/dx=x^2/y from x = 0 to x = 2.4 with y(0)...
Consider the following first-order ODE dy/dx=x^2/y from x = 0 to x = 2.4 with y(0) = 2. (a) solving with Euler’s explicit method using h = 0.6 (b) solving with midpoint method using h = 0.6 (c) solving with classical fourth-order Runge-Kutta method using h = 0.6. Plot the x-y curve according to your solution for both (a) and (b).
Find the eigenvalues of the problem: y'' + λy = 0, 0 < x < 2π,...
Find the eigenvalues of the problem: y'' + λy = 0, 0 < x < 2π, y(0) = y(2π), y' (0) = y'(2π) and one eigenfunction for each eigenvalue.
subject:engineering mathematics solve [cos(x+y)-sin(x+y)]dx-sin(x+y)dy=0,is the ODE exact?Find an integrating factor and solve the ODE.
subject:engineering mathematics solve [cos(x+y)-sin(x+y)]dx-sin(x+y)dy=0,is the ODE exact?Find an integrating factor and solve the ODE.
Let X ~ exp(λ) MGF of X = λ/(1-t) a) What is MGF of Y =...
Let X ~ exp(λ) MGF of X = λ/(1-t) a) What is MGF of Y = 3X b) Y has a common distribution, what is the pdf of Y? c) Let X1,X2,....Xk be independent and identically distributed with Xi ~ exp(λ) and S = Σ Xi (with i = 1 below the summation symbol, and k is on top of the summation symbol). What is the MGF of S? d) S has a common distribution. What is the pdf of...
Determine the eigenvalues and eigenfunctions of the following regular Sturm–Liouville systems: (a) y′′ + λy =...
Determine the eigenvalues and eigenfunctions of the following regular Sturm–Liouville systems: (a) y′′ + λy = 0, y′ (0) = 0 , y′ (π) = 0. (b) y′′ + λy = 0, y (1) = 0 , y(0) + y′ (0) = 0.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT