Question

dy/dx-2y=3ex then find y"​

Answer 1

GIVEN :–

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$\implies \bf \dfrac{dy}{dx} - 2y = 3 {e}^{x}$

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TO FIND :–

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▪︎ Value of 'y' = ?

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SOLUTION :–

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$\implies \bf \dfrac{dy}{dx} - 2y = 3 {e}^{x}$

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• This differential equation is in the form of –

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$\implies \bf \dfrac{dy}{dx} + Py = Q$

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• Now compare –

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$\:\: \longrightarrow \:\: \bf P= - 2$

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$\: \: \longrightarrow \: \: \bf Q= 3 {e}^{x}$

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▪︎ Let's calculate I.F. for linear differential equation –

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$\implies \bf I.F. = {e}^{ \int p.dx}$

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$\implies \bf I.F. = {e}^{ \int -2.dx}$

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$\implies \large{ \boxed { \bf I.F. = {e}^{-2x}}}$

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☯️ SOLUTION for Liner Differential :–

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$\implies \large{ \boxed{ \bf y(I.F.) = \int(Q)(I.F.).dx + c}}$

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• Put the values –

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$\implies \bf y( {e}^{-2x}) = \int(3 {e}^{x})( {e}^{ - 2x} ).dx + c$

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$\implies \bf y( {e}^{-2x}) = \int(3 {e}^{x - 2x}).dx + c$

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$\implies \bf y( {e}^{-2x}) = 3 \int {e}^{-x}.dx + c$

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$\implies \bf y( {e}^{-2x}) = -3 {e}^{-x}+ c$

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$\implies \bf y = -3{e}^{-x}. {e}^{2x} + c. {e}^{2x}$

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$\implies \large { \boxed{ \bf y =-3 {e}^{x}+ c. {e}^{2x}}}$