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Expert-verified Found in: Page 14 ### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069 # Question: In Problems 3–8, determine whether the given function is a solution to the given differential equation.${\mathbf{y}}{\mathbf{=}}{\mathbf{3}}{\mathbf{}}{\mathbf{sin}}{\mathbf{}}{\mathbf{2}}{\mathbf{x}}{\mathbf{+}}{{\mathbf{e}}}^{\mathbf{-}\mathbf{x}}$, ${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{4}}{\mathbf{y}}{\mathbf{=}}{\mathbf{5}}{{\mathbf{e}}}^{\mathbf{-}\mathbf{x}}$

The given function is a solution to the given differential equation.

See the step by step solution

## Step 1: Differentiating the given equation w.r.t. (with respect to) x

Firstly, we will differentiate $\mathrm{y}=3\mathrm{sin}2\mathrm{x}+{\mathrm{e}}^{-\mathrm{x}}$ with respect to x,

$\frac{\mathrm{dy}}{\mathrm{dx}}=6\mathrm{cos}2\mathrm{x}-{\mathrm{e}}^{-\mathrm{x}}$

Again, differentiating the given function with respect to x,

$\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}=-12\mathrm{sin}2\mathrm{x}+{\mathrm{e}}^{-\mathrm{x}}$

## Step 2: Simplification

Putting the values from step 1 in the L.H.S. (Left-hand side) of the given differential equation,

$\mathrm{y}\text{'}\text{'}+4\mathrm{y}=-12\mathrm{sin}2\mathrm{x}+{\mathrm{e}}^{-\mathrm{x}}+4\left(3\mathrm{sin}2\mathrm{x}+{\mathrm{e}}^{-\mathrm{x}}\right)\phantom{\rule{0ex}{0ex}}\mathrm{y}\text{'}\text{'}+4\mathrm{y}=-12\mathrm{sin}2\mathrm{x}+{\mathrm{e}}^{-\mathrm{x}}+12\mathrm{sin}2\mathrm{x}+4{\mathrm{e}}^{-\mathrm{x}}\phantom{\rule{0ex}{0ex}}\mathrm{y}\text{'}\text{'}+4\mathrm{y}=5{\mathrm{e}}^{-\mathrm{x}}$

which is the same as the R.HS. (Right-hand side) of the given differential equation.

Hence, ${\mathbf{y}}{\mathbf{=}}{\mathbf{3}}{\mathbf{}}{\mathbf{sin}}{\mathbf{}}{\mathbf{2}}{\mathbf{x}}{\mathbf{+}}{{\mathbf{e}}}^{\mathbf{-}\mathbf{x}}$ is a solution to the differential equation ${\mathbf{y}}{\mathbf{\text{'}}}{\mathbf{\text{'}}}{\mathbf{+}}{\mathbf{4}}{\mathbf{y}}{\mathbf{=}}{\mathbf{5}}{{\mathbf{e}}}^{\mathbf{-}\mathbf{x}}$.

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