Question

solve it step by step because my answer isn't match to book​

Answer 1

Ans ---> d²y / dx² - m² y = 0

Solution--->

y = A eᵐˣ + B e⁻ᵐˣ .....................(1)

Given equation has two parameters so we differentiate it twice and eliminate parameters to get required differential equation

Differentiating equation ( 1 ) with respect to x , we get

dy / dx = A d/ dx ( eᵐˣ ) + B d / dx ( e⁻ᵐˣ )

= A eᵐˣ d / dx ( mx ) + B e⁻ᵐˣ d / dx ( -mx )

= A eᵐˣ ( m × 1 ) + B e⁻ᵐˣ ( - m × 1 )

= m A eᵐˣ - m B e⁻ᵐˣ

Differentiating it with respect to x again , we get

d²y / dx² = mA d/dx ( eᵐˣ ) - Bm d/dx ( e⁻ᵐˣ )

= mA eᵐˣ d/dx ( mx ) - Bm e⁻ᵐˣ d/dx ( -mx )

= mA eᵐˣ ( m × 1 ) - Bm e⁻ᵐˣ { m × (-1 ) }

= m² A eᵐˣ + m² B e⁻ᵐˣ

= m² ( A eᵐˣ + B e⁻ᵐˣ )

By (1) putting y = Aeᵐˣ + B e⁻ᵐˣ , we get

d²y / dx² = m² y

d²y / dx² - m² y = 0

it is the required differential equation

Formulee used--->

(1) d / dx ( eˣ ) = eˣ

(2) d / dx ( x ) = 1