Why Is y''(a) Determined by y(a) and y'(a) in a Differential Equation?

[AndreaA]

Indicate why we can impose only n initial conditions on a solution of nth order linear differential equation.

A) Given the equation y'' + py'+ qy = 0
explain why the value of y''(a) is determined by the values of y(a) and y'(a).

B) Prove that the equation y'' - 2y' -5y =0
has the solution satisfying the conditions y(0) = 1, y'(0) = 0, and y''(0) = C
if and only if C = 5.

 

[matematikawan]

A) Given the equation y'' + py'+ qy = 0
explain why the value of y''(a) is determined by the values of y(a) and y'(a).

The DE is linear so it must have a general solution which is a linear combination of two linearly independent solutions y₁(x) and y₂(x).

y(x)=c₁y₁(x) + c₂y₂(x) .

c₁ and c₂ can be determined uniquely from the given initial conditions. So the result can be deduce from here.

 

[deluks917]

Do you need to the dimension of the solution set to a second order system is two dimensional?

 

[AlephZero]

A) Given the equation y'' + py'+ qy = 0
explain why the value of y''(a) is determined by the values of y(a) and y'(a).

Because y''(a) = -py'(a) - qy(a).

Move along, please, there's nothing to explain here...