Verify that the function y-aex-be-2x where a-b - R is a solution of the different equation d2ydx2-dydx-2y-

Given, y=ae^x+be^ 2x dydx=ddx(ae^x+be^ 2x) ∴dydx=ae^x 2be^ 2x d^2ydx^2=ddx (dydx #160; )=ddx(ae^x 2be^ 2x) ∴d^2ydx^2=ae^x+4be^ 2x Giv ...

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Standard XII

Mathematics

Summation by Sigma Method

Verify that t...

Question

Verify that the function $y = a e^{x} + b e^{- 2 x}$ where $a, b \in R$ is a solution of the different equation $\frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} = 2 y$ .

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Solution

Given,

$y = a e^{x} + b e^{- 2 x}$

$\frac{d y}{d x} = \frac{d}{d x} (a e^{x} + b e^{- 2 x})$

$∴ \frac{d y}{d x} = a e^{x} - 2 b e^{- 2 x}$

$\frac{d^{2} y}{d x^{2}} = \frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (a e^{x} - 2 b e^{- 2 x})$

$∴ \frac{d^{2} y}{d x^{2}} = a e^{x} + 4 b e^{- 2 x}$

Given,

$\frac{d^{2} y}{d x^{2}} + \frac{d y}{d x}$

$= a e^{x} + 4 b e^{- 2 x} + a e^{x} - 2 b e^{- 2 x}$

$= 2 a e^{x} + 2 b e^{- 2 x}$

$= 2 (a e^{x} + b e^{- 2 x})$

$= 2 y$

$∴ \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} = 2 y$

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Similar questions

Q. Show that

y = a e^{2 x} + b e^{- x}

is a solution of the differential equation

\frac{d^{2} y}{d x^{2}} - \frac{d y}{d x} - 2 y = 0

Q. Verify

y = a e^{x}

is a solution of

\frac{d^{2} y}{d x^{2}} - y = 0

, find particular solution when

y (0) = 1

Q. Prove that

y = {a e}^{- 2 x} + {b e}^{x}

is the solution of differential equation

\frac{d^{2} y}{{d x}^{2}} + \frac{d y}{d x} - 2 y = 0

Q. The differential equation of the family of curves

y = e^{x} (A cos x + B sin x)

where

A, B

are arbitary constants is

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i)

(ii)

(iii)

(iv)

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Verify that the function y=aex+be−2x where a,b∈R is a solution of the different equation d2ydx2+dydx=2y.

Given,y=aex+be−2xdydx=ddx(aex+be−2x)∴dydx=aex−2be−2xd2ydx2=ddx(dydx)=ddx(aex−2be−2x)∴d2ydx2=aex+4be−2xGiven,d2ydx2+dydx=aex+4be−2x+aex−2be−2x=2aex+2be−2x=2(aex+be−2x)=2y∴d2ydx2+dydx=2y

Verify that the function $y = a e^{x} + b e^{- 2 x}$ where $a, b \in R$ is a solution of the different equation $\frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} = 2 y$ .