Question

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

$\mathbf{y}\text{'}\text{'}\text{'}\mathbf{+}\mathbf{2}\mathbf{y}\text{'}\text{'}\mathbf{+}\mathbf{y}\text{'}\mathbf{=}\mathbf{10}$

Short answer

$C_1+C_2{e}^{-x}+C_{3}x{e}^{-x}+10x$

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.

Solve the differential equation by the undetermined coefficient method

Consider the following differential equation:

$y\text{'}\text{'}\text{'}+2y\text{'}\text{'}+y\text{'}=10$

The auxiliary equation is,

$\begin{array}{rcl}{m}^3+2m^2+m& =& 0\\ m\left(m^2+2m+1\right)& =& 0\\ m(m+1{)}^2& =& 0\\ m& =& 0,-1,-1\end{array}$

Thus, the complementary solution of the differential equation (1) is,$y_c=C_1+C_2{e}^{-x}+C_{3}x{e}^{-x}.$

Find the particular integral

Since$D\left(10\right)=0$

So, apply the differential operator D to the both sides of given equation, to get

$\begin{array}{rcl}D\left(D^3+2D^2+D\right)y& =& D\left(10\right)\\ & =& 0\end{array}$

So, the auxiliary equation corresponding to the above equation is,

$\begin{array}{rcl}m\left(m^3+2m^2+m\right)& =& 0\\ m^2\left(m^2+2m+1\right)& =& 0\\ m^2(m+1{)}^2& =& 0\\ m_1& =& 0=m_2,m_3=-1=m_4\end{array}$

Thus, the general solution must be $y=\underset{y_c}{\underbrace{C_1+C_2{e}^{-x}+C_{3}x{e}^{-x}}}+C_{4}x$.

Substitution

The general solution of the differential equation of the form$y=y_c+y_p$ .

So, the particular solution should be of the form $y_p=Ax$.

$y_p\text{'}=A,y_p\text{'}\text{'}=0,y_p\text{'}\text{'}\text{'}=0.$

Substitute the above values in.$y\text{'}\text{'}\text{'}+2y\text{'}\text{'}+y\text{'}=10$

$\begin{array}{rcl}0+2\left(0\right)+A& =& 10\\ A& =& 10\end{array}$

Substitute the value$A=10$ in$y_p=Ax$

Thus,$y_p=10x$

Thus, the general solution of the given differential equation is,

$\begin{array}{rcl}y& =& y_c+y_p.\\ & =& C_1+C_2{e}^{-x}+C_{3}x{e}^{-x}+10x\end{array}$

Hence, the general solution is$C_1+C_2{e}^{-x}+C_{3}x{e}^{-x}+10x$