If y = 3e^ 2x + 2e^ 3x , prove that d^ 2 y d x^ 2 -5 d y d x + 6y… - Vidyadip

Question

If $y = 3e^{2x} + 2e^{3x} ,$ prove that $\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x} + 6y = 0.$

✓

Answer

Given that $y = 3e^{2x} + 2e^{3x},$
Then $\frac{d y}{d x} = 6e^{2x} + 6e^{3x}$
$= 6 (e^{2x} + e^{3x})$
Therefore, $\frac{d^{2} y}{d x^{2}} = 12e^{2x} + 18e^{3x}$
$= 6 (2e^{2x} + 3e^{3x})$
Hence $\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x} + 6y = 6 (2e^{2x} + 3e^{3x}) – 30 (e^{2x} + e^{3x}) + 6 (3e^{2x} + 2e^{3x}) = 0.$
So, the result is correct.

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