One solution of the differential equation

e1s2kat26
2021-01-28
Answered

One solution of the differential equation

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krolaniaN

Answered 2021-01-29
Author has **86** answers

Given differential equation is ${y}^{\u2033}-4y=0$ (1)

Given that First solution is${y}_{1}={e}^{2x}$

let general solution$y=v\cdot {y}_{1}$

substitute${y}_{1}={e}^{2x}$

$y=v{e}^{2x}$

Differentiate$y=v{e}^{2x}$ with respect to x

${y}^{\prime}=\frac{d}{dx}(v{e}^{2x})$

$=\frac{vd}{dx}{e}^{2x}+{e}^{2x}\frac{d}{dx}(v)$

$=v\cdot 2{e}^{2x}+{e}^{2x}\cdot {v}^{\prime}$

$=2v{e}^{2x}+{e}^{2x}{v}^{\prime}$

hence,${y}^{\prime}=2v{e}^{2x}+{e}^{2x}v,$

Again differentiate${y}^{\prime}=2v{e}^{2x}+{e}^{2x}{v}^{\prime}$ with respect to x

${y}^{\u2033}=\frac{d}{dx(2v{e}^{2x}+{e}^{2x}{v}^{\prime})}$

$=2\frac{d}{dx}(v{e}^{2x})+\frac{d}{dx}({e}^{2x}{v}^{\prime})$

$=4v{e}^{2x}+2{e}^{2x}{v}^{\prime}+2{e}^{2x}{v}^{\prime}+{e}^{2x}{v}^{\u2033}$

$=4v{e}^{2x}+4{e}^{2x}{v}^{\prime}+{e}^{2x}{v}^{\u2033}$

Hence,${y}^{\u2033}=4v{e}^{2x}+4{e}^{2x}{v}^{\prime}+{e}^{2x}{v}^{\u2033}$

Now, substitute${y}^{\u2033}=4v{e}^{2x}+4{e}^{2x}{v}^{\prime}+{e}^{2x}{v}^{\u2033}$ and $y=v{e}^{2x}$ in equation (1) and simlify it

$4v{e}^{2x}+4{e}^{2x}{v}^{\prime}+{e}^{2x}{v}^{\u2033}-4v{e}^{2x}=0$

$4{e}^{2x}{v}^{\prime}+{e}^{2x}{v}^{\u2033}=0$ (2)

Let$w={v}^{\prime}$

$\Rightarrow {w}^{\prime}={v}^{\u2033}$

Substitute these value in equation (2)

$4{e}^{2x}w+{e}^{2x}{w}^{\prime}=0$

${e}^{2x}(4w+{w}^{\prime})=0$ (Take common as ${e}^{2x}$ )

Divide both side be${e}^{2x}$ and simplify it

$\frac{{e}^{2x}(4w+{w}^{\prime})}{{e}^{2x}}=\frac{0}{{e}^{2x}}$

$4w+{w}^{\prime}=0$

Substract 4w from both sides and simplify it

$4w-4w+{w}^{\prime}=-4w$

${w}^{\prime}=-4w$

$\frac{dw}{dx}=-4w$

$dww=-4dx$

Now integrate it

$\int \frac{dw}{w}=-4\int dx+c$

$\mathrm{ln}w=-4x+c$

$w={e}^{-4x+c}$

$w={e}^{-4x}\cdot {e}^{c}$

Further simplify it

$w={c}_{1}{e}^{-4x}$ (Since

Given that First solution is

let general solution

substitute

Differentiate

hence,

Again differentiate

Hence,

Now, substitute

Let

Substitute these value in equation (2)

Divide both side be

Substract 4w from both sides and simplify it

Now integrate it

Further simplify it

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