Question

In Problems 35–64 solve the given differential equation by undetermined coefficients.

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

Short answer

$y=C_1{e}^x+C_{2}x{e}^x+32+22x+6x^2+x^3$

Step-by-step solution

Definition of homogeneous differential equation

A homogeneous differential equation is an equation containing a differentiation and a function, with a set of variables.

Given data

From$m^2-2m+1=0$we find$m_1=1=m_2$

So, the complementary function is $y_c=C_1{e}^x+C_{2}x{e}^x$.

Apply differential operator

Since $D^4\left(x^3+4x\right)=0$, We apply the differential operator$D^4$ , to the both sides of the given equation to get

$\begin{array}{rcl}{D}^4\left(D^2-2D+1\right)y& =& D^4\left(x^3+4x\right)\\ D^4\left(D^2-2D+1\right)y& =& 0\end{array}$

now, the auxiliary equation is

$\begin{array}{rcl}{m}^4\left(m^2-2m+1\right)& =& 0\\ m^4(m-1{)}^2& =& 0\\ & \Rightarrow & m_1=1=m_2,m_3=0=m_4=m_5=m_6\end{array}$

Thus, the general solution is $y=\underset{y_c}{\underbrace{C_1{e}^x+C_{2}x{e}^x+C_3}}+C_{4}x+C_5{x}^2+C_6{x}^3$.

Substitution

Particular solution, $y_p$

$y_p=C_3+C_{4}x+C_5{x}^2+C_6{x}^3$

Then

$$\begin{gathered} y_p\text{'}=C_4+2C_{5}x+3C_6{x}^2 \\ {\dot{y}}_p\text{'}\text{'}=2C_5+6C_{6}x \end{gathered}$$

Now, substituting these into the given differential equation gives

$\begin{array}{rcl}2C_5+6C_{6}x-2C_4-4C_{5}x-6C_6{x}^2+C_3+C_{4}x+C_5{x}^2+C_6{x}^3& =& x^3+4x\\ C_6{x}^3+\left(C_5-6C_6\right)x^2+\left(6C_6-4C_5+C_4\right)x+\left(2C_5-2C_4+C_3\right)& =& x^3+4x\end{array}$

Equating the coefficient

On equating the coefficients of like terms, we get

$\begin{array}{rcl}{C}_6& =& 1\\ C_5-6C_6& =& 0\\ & \Rightarrow & C_5=6\end{array}$

$\begin{array}{rcl}6C_6-4C_5+C_4& =& 4\\ & \Rightarrow & C_4=22\end{array}$

$\begin{array}{rcl}2C_5-2C_4+C_3& =& 0\\ & \Rightarrow & C_3=32\end{array}$

So, $y_p=32+22x+6x^2+x^3$

Thus, the general solution is $y=C_1{e}^x+C_{2}x{e}^x+32+22x+6x^2+x^3$.