Question

In Problems 1–26 solve the given differential equation by undetermined coefficients.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{-}\mathbf{3}$

Short answer

$y=k_2{e}^x+3x+k_1$

Step-by-step solution

Finding the corresponding homogeneous equation

We are given with the non-homogeneous differential equation $y\text{'}\text{'}-y\text{'}=-3$. Consider$y\text{'}\text{'}-y\text{'}=-3$as the solution of the differential equation,

Substitute $y=e^{mx},y\text{'}=m{e}^{mx},y\text{'}\text{'}=m^2{e}^{mx}$, into $y\text{'}\text{'}-y\text{'}=0$to obtain the auxiliary equation.

role="math" localid="1663830634993" $\begin{array}{c}{\mathbf{m}}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{m}\mathbf{x}}\mathbf{-}\mathbf{m}{\mathbf{e}}^{\mathbf{m}\mathbf{x}}\mathbf{=}\mathbf{0}\\ {\mathbf{e}}^{\mathbf{m}\mathbf{x}}\left(m^2-m\right)\mathbf{=}\mathbf{0}\end{array}$

Since $e^{mx}$cannot be equal to 0, then we have

$\begin{array}{c}{m}^2-m=0\\ m\left(m-1\right)=0\end{array}$

Then the roots are which are $m_1=0,m_2=1$complex roots.

Then the solution of the corresponding homogeneous differential equation $y\text{'}\text{'}-y\text{'}=0$can be given by

$\begin{array}{c}{y}_h=k_1{e}^{m_{1}x}+k_2{e}^{m_{2}x}\\ =k_1{e}^0+k_2{e}^x\\ =k_1+k_2{e}^x\end{array}$

Finding the general solution for the non-homogeneous differential equation

Secondly, we need to find the particular solution of the non-homogeneous differential equation$y\text{'}\text{'}-y\text{'}=-3$

Assume that$y_p=Ax$ is the solution for the non-homogeneous differential equation where$g\left(x\right)=-3$ , since we can have a free constant in the homogeneous part.

Differentiate the assumption with respect to x, we can obtain,

$y_p\text{'}=A\mathrm{and}{y}_p\text{'}\text{'}=0$

Substituteinto the differential equation, then we can obtain,

$\begin{array}{c}\left(0\right)-\left(A\right)=-3\\ A=3\end{array}$

Then the assumed solution becomes $y_p=3x$.

From the equations, we can obtain,

$y=k_2{e}^x+3x+k_1$

Thus the general solution of the given differential equation is $y=k_2{e}^x+3x+k_1$.