Question

Consider the differential equation (y ′′)² −y² =0.

(a) Use a substitution method to reduce this ODE to four separate first order separable ODEs in y
(b) Check that eˣ,e −ˣ ,cos(x) and sin(x) are all solutions to the ODE, and that y= c₁​ cos(x)+c₂ sin(x) and y=c₁​ eˣ +c₂​e −x are solutions, but y=c₁​ eˣ +c₂​ sin(x) is not a solution. Try to explain why that is the case. ³

Answer 1

By using substitution methods and verification through direct substitution, we can reduce the given second-order differential equation to first-order ones and also confirm that specific exponential and trigonometric functions are solutions to the equation.

Step-by-step explanation:

The differential equation given is (y'')2 − y2 = 0. To solve this using substitution, we first let v = y'', which leads to v2 = y2. We can then take the square root of both sides, giving us two possible equations: v = y and v = -y. If we let u = y', we can translate these into first-order differential equations: u' = y for v = y and u' = -y for v = -y.

To verify the solutions to the original equation, we consider the provided functions: ex, e-x, cos(x), and sin(x). We can show that each, when plugged into the original differential equation, satisfies the relationship (derivative of the function)2 = (the function itself)2. Combinations of these functions also work, specifically y = c1 cos(x) + c2 sin(x) and y = c1 ex + c2 e-x, as their second derivatives still adhere to the required relationship.

However, the combined function y = c1 ex + c2 sin(x) is not a solution because its second derivative does not produce the necessary equality.