If y=50e^10x + 60e^-10x,prove that d^2y/dx^2=100 y
Answers
Given : y = 50e¹⁰ˣ + 60e⁻¹⁰ˣ
To find : Prove that d²y/dx² = 100y
Solution:
y = 50e¹⁰ˣ + 60e⁻¹⁰ˣ
Differentiating wrt x
dy/dx = d (50e¹⁰ˣ + 60e⁻¹⁰ˣ)/dx
using d(A + B)/dx = dA/dx + dB/dx and dCf(x)/dx = Cd(f(x)/dx)
dy/dx = 50d(e¹⁰ˣ/dx) + 60d (e⁻¹⁰ˣ)/dx
=> dy/dx = 50 * 10e¹⁰ˣ + 60 (-10)e⁻¹⁰ˣ
=> dy/dx = 500e¹⁰ˣ - 600 e¹⁰ˣ
Differentiating wrt x again
=> d²y/dx² = 500 *d (10e¹⁰ˣ)/dx -600 d(e⁻¹⁰ˣ )/dx
=> d²y/dx² = 500 * (10e¹⁰ˣ) - 600(-10e⁻¹⁰ˣ)
=> d²y/dx² = 5000e¹⁰ˣ + 600 0e⁻¹⁰ˣ
taking 100 common
=> d²y/dx² = 100 (50e¹⁰ˣ + 60e⁻¹⁰ˣ)
=> d²y/dx² = 100y
QED
Hence Proved
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