Question

Solve the given differential equation by undetermined coefficients.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{5}\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{4}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathit{x}\mathbf{+}\mathbf{6}$

Short answer

The general solution of the given differential equation is

$y=-\frac{2}{5}{x}^4+\frac{14}{75}{x}^3+\frac{53}{250}{x}^2+\frac{697}{625}x+k_1+k_2{e}^{5x}$

Step-by-step solution

Note the given data

Given the second order differential equation $y\text{'}\text{'}-5y\text{'}=2x^3-4x^2-x+6$

Obtain the general solution of the homogeneous differential equation  

We know a homogeneous differential equation is an equation containing

differentiation and a function with a set of variables.

First, we need to solve the associated homogeneous equation$y\text{'}\text{'}-5y\text{'}=0$

Corresponding auxiliary equation is$m^2-5m=0$

Solving,

$m(m-5)=0$

m₁ =0 and m₂ = 5, which are real and distinct.

So the solution of corresponding homogeneous equation is

$y_h=k_1{e}^{0x}+k_2{e}^{5x}=k_1+k_2{e}^{5x}$

Finding the particular solution

Given non homogeneous differential equation is$y\text{'}\text{'}-5y\text{'}=2x^3-4x^2-x+6$

Assume that $y_p=(A{x}^4+B{x}^3+C{x}^2+Dx)$is a solution of the non-homogeneous differential equation

Differentiate with respect to x

$y_p\text{'}=(4A{x}^3+3B{x}^2+2Cx+D)$……(1)

$y_p\text{'}\text{'}=(12A{x}^2+6Bx+2C)$……(2)

Substituting (1) and (2) in given differential equation

$12A{x}^2+6Bx+2C)-5(4A{x}^3+3B{x}^2+2Cx+D)=2x^3-4x^2-x+6$

Or$-20A{x}^3+(12A-15B)x^2+(6B-10C)x+(2C+5D)=2x^3-4x^2-x+6$

comparing the coefficients,

$-20A=2\Rightarrow A=\frac{-1}{10}$

$12A-15B=-4\Rightarrow B=\frac{14}{75}$

$6B-10C=-1\Rightarrow C=\frac{53}{250}$

$2C+5D=6\Rightarrow D=\frac{697}{625}$

Hence the particular solution is (1) becomes

$y_p=\frac{-2}{5}{x}^4+\frac{14}{75}{x}^3+\frac{53}{250}{x}^2+\frac{697}{625}x$

Finding the solution

Hence the solution of given differential equation is

y=y_h+y_p

$y=-\frac{2}{5}{x}^4+\frac{14}{75}{x}^3+\frac{53}{250}{x}^2+\frac{697}{625}x+k_1+k_2{e}^{5x}$