Find y as a function of x if :

Question

Find y as a function of x if :

y′′′−5y′′−y′+5y=0,

y(0)=−4,  y′(0)=−9,  y′′(0)=92.

y(x)= ____________

Answer 1

So with this third order linear differential equation (which happens to have constant coefficients), we want to see if we can determine a solution with mimics y = e^rx. Here r is a parameter we wish to solve, thereby changing the differential equation into an algebraic one...

y(x) = e^rx

y'(x) = re^rx

y"(x) = r²e^rx

y"'(x) = r³e^rx

Plugging in our ansatz (guess solution)...

r³e^rx - 5r²e^rx - re^rx + 5e^rx = 0

e^rx (r³ - 5r² - r + 5) = 0

We can safely divide by e^rx since it cannot be zero unless r*x = +/- ∞

Therefore, we are looking for solutions of the cubic equation...

r³ - 5r² - r + 5 = 0

(r³ - 5r²)- (r - 5) = 0

r²(r - 5) - (r - 5) = 0

(r² - 1) * (r - 5) = 0

This leads to the following solutions for r...

r = 5, -1, +1

Therefore the homogeneous (null/complimentary) solution is

y(x) = Ae^5x + Be^-x + Ce^x

Using the initial values we can ascertain A,B,C

y'(x) = 5Ae^5x - Be^-x + Ce^x

y"(x) = 25Ae^5x + Be^-x + Ce^x

Plugging in x = 0 for each of these derivatives...

y(0) = -4 = A + B + C

y'(0) = -9 = 5A - B + C

y"(0) = 92 = 25A + B + C

You can use linear algebra or substitution methods to solve this system of equations...

A = 4

B = 21/2

C = -37/2

y(x) = 4e^5x + 21/2 * e^-x - 37/2 * e^x