Question

Consider the differential equation$\mathit{a}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathit{b}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathit{c}\mathit{y}\mathbf{=}\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}$, where,a, b and care constants. Choose the input functions$\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}$forwhich the method of undetermined coefficients is applicableand the input functions for which the method of variation ofparameters is applicable.

(a)role="math" localid="1663898381084" $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{e}}^{\mathbf{x}}\mathit{l}\mathit{n}\mathit{x}$(b)role="math" localid="1663898398048" $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{3}}\mathit{c}\mathit{o}\mathit{s}\mathit{x}$(c)role="math" localid="1663898412823" $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{x}}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$

(d)role="math" localid="1663898362328" $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{-}\mathbf{2}}{\mathit{e}}^{\mathbf{x}}$(e)role="math" localid="1663898345398" $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{x}$(f)$\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{x}}}{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{x}}$

Short answer

The input functions which are applicable for method of undetermined coefficients are ${\text{g(x)=e}}^{\text{x}}{\text{lnx,g(x)=x}}^{\text{3}}{\text{cosx,g(x)=e}}^{\text{-x}}{\text{sinx,g(x)=2x}}^{\text{-2}}{\text{e}}^{\text{x}}\text{,g(x)=}\frac{{\text{e}}^{\text{x}}}{\text{sinx}}$and for method of variation of parameters are ${\text{g(x)=x}}^{\text{3}}{\text{cosx,g(x)=e}}^{\text{-x}}{\text{sinx,g(x)=sin}}^{\text{2}}\text{x}$.

Step-by-step solution

Step 1:DefineCauchy-Euler equation.

The method of indeterminate coefficients is a method for finding a specific solution to nonhomogeneous ordinary differential equations and recurrence relations in mathematics.

The variation of parameters is a generic approach for identifying a specific solution to a differential equation by substituting functions for the constants in the solution of a related (homogeneous) equation and calculating these functions so that the original differential equation is satisfied.

Find the input functions for which the method of undetermined coefficient is applicable.

Let the second order differential equation be,$\text{ay''+by'+cy=g(x)}$ .

The input functions $\text{g(x)}$which are applicable for method of undetermined coefficients are,

$\begin{array}{l}{\text{a)g(x)=e}}^{\text{x}}\text{lnx}\\ {\text{b)g(x)=x}}^{\text{3}}\text{cosx}\\ {\text{c)g(x)=e}}^{\text{-x}}\text{sinx}\\ {\text{d)g(x)=2x}}^{\text{-2}}{\text{e}}^{\text{x}}\\ \text{f)g(x)=}\frac{{\text{e}}^{\text{x}}}{\text{sinx}}\end{array}$

Find the input functions for which the method of undetermined coefficient is applicable.

The input functions which are applicable for method of variation of parameters are,$\begin{array}{l}{\text{b)g(x)=x}}^{\text{3}}\text{cosx}\\ {\text{c)g(x)=e}}^{\text{-x}}\text{sinx}\\ {\text{e)g(x)=sin}}^{\text{2}}\text{x}\end{array}$