Question

Suppose${\mathit{m}}_{\mathbf{1}}\mathbf{=}\mathbf{3}\mathbf{,}{\mathit{m}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{5}\mathbf{,}$and${\mathit{m}}_{\mathbf{3}}\mathbf{=}\mathbf{1}$are roots of multiplicityone, two, and three, respectively, of an auxiliary equation. Writedown the general solution of the corresponding homogeneouslinear DE if it is

(a) an equation with constant coefficients,

(b) a Cauchy-Euler equation

Short answer

The general solution of the corresponding homogeneous linear differential equation if it is,

(a) An equation with constant coefficients is.${\text{y=c}}_{\text{1}}{\text{e}}^{\text{3x}}{\text{+c}}_{\text{2}}{\text{e}}^{\text{-5x}}{\text{+c}}_{\text{3}}{\text{xe}}^{\text{-5x}}{\text{+c}}_{\text{4}}{\text{e}}^{\text{x}}{\text{+c}}_{\text{5}}{\text{xe}}^{\text{x}}{\text{+c}}_{\text{6}}{\text{x}}^{\text{2}}{\text{e}}^{\text{x}}$

(b) a Cauchy-Euler equation is.${\text{y=c}}_{\text{1}}{\text{x}}^{\text{3}}{\text{+c}}_{\text{2}}{\text{x}}^{\text{-5}}{\text{+c}}_{\text{3}}{\text{x}}^{\text{-5}}{\text{lnx+c}}_{\text{4}}{\text{x+c}}_{\text{5}}{\text{xlnx+c}}_{\text{6}}{\text{x(lnx)}}^{\text{2}}$

Step-by-step solution

Step 1:DefineCauchy-Euler equation.

A linear homogeneous ordinary differential equation with variable coefficients is known as a Euler–Cauchy equation, Cauchy–Euler equation, or simply Euler's equation in mathematics.

Find the differential equation.

Since the differential equation has the roots , $\text{m=3,-5,1}$then the auxiliary equation of the differential equation is ${\text{(m-3)(m+5)}}^{\text{2}}{\text{(m+1)}}^{\text{3}}\text{=0}$.

The general solution of the differential equation of the required two cases are as follows:

(a) For constant coefficients,

$\begin{array}{c}{\text{y=c}}_{\text{1}}{\text{e}}^{{\text{m}}_{\text{1}}\text{x}}{\text{+c}}_{\text{2}}{\text{e}}^{{\text{m}}_{\text{2}}\text{x}}{\text{+c}}_{\text{3}}{\text{xe}}^{{\text{m}}_{\text{2}}\text{x}}{\text{+c}}_{\text{4}}{\text{e}}^{{\text{m}}_{\text{3}}\text{x}}{\text{+c}}_{\text{5}}{\text{xe}}^{{\text{m}}_{\text{3}}\text{x}}{\text{+c}}_{\text{6}}{\text{x}}^{\text{2}}{\text{e}}^{{\text{m}}_{\text{3}}\text{x}}\\ {\text{=c}}_{\text{1}}{\text{e}}^{\text{3x}}{\text{+c}}_{\text{2}}{\text{e}}^{\text{-5x}}{\text{+c}}_{\text{3}}{\text{xe}}^{\text{-5x}}{\text{+c}}_{\text{4}}{\text{e}}^{\text{x}}{\text{+c}}_{\text{5}}{\text{xe}}^{\text{x}}{\text{+c}}_{\text{6}}{\text{x}}^{\text{2}}{\text{e}}^{\text{x}}\end{array}$

(b) For Cauchy Euler equation,

$\begin{array}{c}{\text{y=c}}_{\text{1}}{\text{x}}^{{\text{m}}_{\text{1}}}{\text{+c}}_{\text{2}}{\text{x}}^{{\text{m}}_{\text{2}}}{\text{+c}}_{\text{3}}{\text{x}}^{{\text{m}}_{\text{2}}}{\text{lnx+c}}_{\text{4}}{\text{x}}^{{\text{m}}_{\text{3}}}{\text{+c}}_{\text{5}}{\text{x}}^{{\text{m}}_{\text{3}}}{\text{lnx+c}}_{\text{6}}{\text{x}}^{{\text{m}}_{\text{3}}}{\text{(lnx)}}^{\text{2}}\\ {\text{=c}}_{\text{1}}{\text{x}}^{\text{3}}{\text{+c}}_{\text{2}}{\text{x}}^{\text{-5}}{\text{+c}}_{\text{3}}{\text{x}}^{\text{-5}}{\text{lnx+c}}_{\text{4}}{\text{x+c}}_{\text{5}}{\text{xlnx+c}}_{\text{6}}{\text{x(lnx)}}^{\text{2}}\end{array}$