Infinite differential equation

Calculus Level 4

$\large y = \sum_{n=1}^\infty \dfrac{d^n y}{dx^n}$

If the general solution to the differential equation above can be expressed as

$\alpha (\sin x - \cos x) + Ce^{ \beta x}.$

Find the value of $\alpha + \beta$ .

Note: $C$ is an integration constant.

Clarification : $\dfrac{d^n y}{dx^n}$ denotes the $n^\text{th}$ derivative of $y$ with respect to $x$ .

The answer is 0.5.

1 solution

Samrat Mukhopadhyay
Sep 15, 2016

Note that the given equation can be simplified to obtain, $2\frac{dy}{dx}=y\implies y=Ae^{x/2}$ Hence the solution is $\boxed{0.5}$ .

Could you please explain more @Samrat Mukhopadhyay

Muhammad Ahmad - 4 years, 6 months ago

Write $y_n=\frac{d^ny}{dx^n}$ , then the given equation says $y=\sum_{n\ge 1} y_n$ . Differentiating both sides, we get $y_1=\sum_{n\ge 2}y_n=y-y_1\implies 2\frac{dy}{dx}=y$

Samrat Mukhopadhyay - 4 years, 6 months ago

Thanks alot @Samrat Mukhopadhyay

Muhammad Ahmad - 4 years, 6 months ago

Infinite differential equation

The answer is 0.5.

1 solution

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