Question

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

$\mathbf{y}\text{'}\text{'}\mathbf{+}\mathbf{y}\text{'}\mathbf{=}\mathbf{3}$

Short answer

$y=C+C_2{e}^{-x}+3x$

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.

Find the complementary function

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

So, the complementary function is $y_c=C+C_2{e}^{-x}$.

Apply differential operator

Now, since$D\left(3\right)=0$ , we apply the differential operator D to both sides of given equation, to get

$D\left(D^2+D\right)y=D\left(3\right)$

i.e,$D\left(D^2+D\right)y=0$

So, the auxiliary equation corresponding to the above equation is$m\left(m^2+m\right)=0$

$\Rightarrow m_1=0,m_2=-1,m_3=0$

So, the general solution must be $y=C_1+C_2{e}^{-x}+C_{3}x$.

Substitution

Since $y=y_c+y_p;y_p$should of the form:

$y_p=Ax$

Substituting into the given differential equation gives

$A=3$

$\therefore y_p=3x$

Thus, the general solution is $y=C+C_2{e}^{-x}+3x$.