Second plus order ode formulas reference

Homogeneous Equations

Characteristic Equations

$$\begin{array}{rlrlr}a{y}^{\prime \prime }+b{y}^{\prime }+cy=f(t)& & \text{OR}& & a\frac{d^{2}y}{d{t}^2}+b\frac{dy}{dt}+cy=f(t)\\ \\ & & f(t)=0\end{array}$$

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$$y=e^{rt}$$

the derivatives of $e^{rt}$ are just constants times $e^{rt}$ and the solution must have the property that its second derivative can be expressed as a linear combination of its first and zeroth derivatives.

$$a{r}^2{e}^{rt}+br{e}^{rt}+c{e}^{rt}=0$$

$$\Rightarrow$$

$$e^{rt}(a{r}^2+br+c)=0$$

There is no value you can substitute t for in $e^{rt}$ to get 0, so we can divide both sides by $e^{rt}$

$$\begin{array}{r}a{r}^2+br+c=0\end{array}$$

solve for the roots ($r_n$) using any method (quadratic formula, factoring, etc...)

$$\begin{array}{rlrlr}{r}_1=\frac{-b+\sqrt{b^2-4ac}}{2a}& & \text{And}& & r_2=\frac{-b-\sqrt{b^2-4ac}}{2a}\end{array}$$

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General Solutions to Characteristic Equations

• Quadratics

• 2 Real, Unique Roots

$$\begin{array}{r}y(t)=c_1{e}^{r_{1}t}+c_2{e}^{r_{2}t}\end{array}$$

• double root

$$\begin{array}{r}y(t)=c_1{e}^{rt}+c_2{e}^{rt}t\end{array}$$

• 2 Complex Roots

$$r=\alpha \pm \beta i$$

$$\begin{array}{r}y(t)=c_1{e}^{\alpha t}\cos (\beta t)+c_2{e}^{\alpha t}\sin (\beta t)\end{array}$$

• Cubic

• 3 Real, Unique Roots

$$\begin{array}{r}y(t)=c_1{e}^{r_{1}t}+c_2{e}^{r_{2}t}+c_3{e}^{r_{3}t}\end{array}$$

• 3 Real Roots, 1 Double Root

$$\begin{array}{r}y(t)=c_1{e}^{r_{1}t}+c_2{e}^{rt}+c_3{e}^{rt}t\end{array}$$

• 1 Real Root, 2 Complex Roots

$$\begin{array}{r}y(t)=c_1{e}^{rt}+c_2{e}^{\alpha t}\cos (\beta t)+c_3{e}^{\alpha t}\sin (\beta t)\end{array}$$

Cauchy-Euler Equations

$$\begin{array}{r}a{t}^2{y}^{\prime \prime }+bt{y}^{\prime }+cy=0\\ \\ \text{OR}\\ \\ y^{\prime \prime }+\frac{b}{at}{y}^{\prime }+\frac{c}{a{t}^2}=0\end{array}$$

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assuming that $t>0$

$$\begin{array}{rl}& y(t)=t^r\\ \\ & \frac{dy}{dt}=r{t}^{r-1}\\ \\ & \frac{d^{2}y}{d{t}^2}=(r^2-r)t^{r-2}\\ \\ & \text{plugging into the original formula}\end{array}$$

$$\begin{array}{r}a{t}^2(r^2-r)t^{r-2}+bt(r)t^{r-1}+c{t}^r=0\\ \end{array}$$

$$\begin{array}{r}\frac{a{t}^2(r^2-r)t^r}{t^2}+\frac{bt(r)t^r}{t}+c{t}^r=0\$a(r^2-r)t^r+b(r)t^r+c{t}^r=0\end{array}$$

$$t^r(a{r}^2-ar+br+c)=0\Rightarrow a{r}^2-ar+br+c=0$$

$$a{r}^2+(b-a)r+c=0$$

General Solutions to Cauchy-Euler Equations

Real, unique roots

$$y(t)=c_1{t}^{r_1}+c_2{t}^{r_2}$$

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Double root

$$y(t)=c_1{t}^r+c_2{t}^r\ln (t)$$

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imaginary roots

$$r=\alpha \pm \beta i$$

$$y(t)=c_1{t}^{\alpha }\cos (\beta \ln (t))+c_2{t}^{\alpha }\sin (\beta \ln (t))$$

Non Homogeneous Equations

Particular, Homogeneous, and General Solutions

$$\begin{array}{rlrlr}a{y}^{\prime \prime }+b{y}^{\prime }+cy=f(t)& & \text{OR}& & a\frac{d^{2}y}{d{t}^2}+b\frac{dy}{dt}+cy=f(t)\end{array}$$

$$f(t)\ne 0$$

Homogeneous Solution

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=0\Rightarrow y_c(t)$$

Non Homogeneous Solution

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=f(t)\Rightarrow y_p(t)$$

General Solution

$$y(t)=y_p(t)+y_c(t)$$

Linear Equations (Constant Coefficients)

Undetermined Coefficients

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=C{t}^m{e}^{rt}$$

where $m$ is a nonnegative integer, use the form

$$y_p(t)=t^s(A_m{t}^m+\cdots +A_{1}t+A_0)e^{rt}$$

• $s=0$ if $r$ is not a root of the associated auxiliary equation
• $s=1$ if $r$ is a simple root of the associated auxiliary equation
• $s=2$ if $r$ is a double root of the associated auxiliary equation

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$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=\{\begin{array}{c}C{t}^m{e}^{\alpha t}\cos \beta t\\ \text{ or }\\ C{t}^m{e}^{\alpha t}\sin \beta t\end{array}$$

for $\beta \ne 0$, use the form

$$\begin{array}{rl}{y}_p(t)=& t^s(A_m{t}^m+\cdots +A_{1}t+A_0)e^{\alpha t}\cos \beta t\\ & +t^s(B_m{t}^m+\cdots +B_{1}t+B_0)e^{\alpha t}\sin \beta t,\end{array}$$

• $s=0$ if $\alpha +\beta i$ is not a root of the associated auxiliary equation; and
• $s=1$ if $\alpha +\beta i$ is a root of the associated auxiliary equation.

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$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=P_m(t)e^{rt},$$

where $P_m(t)$ is a polynomial of degree $m$, use the form

$$y_p(t)=t^s(A_m{t}^m+\cdots +A_{1}t+A_0)e^{rt};$$

• $s=0$ if $r$ is not a root of the associated auxiliary equation
• $s=1$ if $r$ is a simple root of the associated auxiliary equation
• $s=2$ if $r$ is a double root of the associated auxiliary equation

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$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=P_m(t)e^{\alpha t}\cos \beta t+Q_n(t)e^{\alpha t}\sin \beta t,\beta \ne 0,$$

where $P_m(t)$ is a polynomial of degree $m$ and $Q_n(t)$ is a polynomial of degree $n$, use the form

$$\begin{array}{rl}{y}_p(t)=& t^s(A_k{t}^k+\cdots +A_{1}t+A_0)e^{\alpha t}\cos \beta t\\ & +t^s(B_k{t}^k+\cdots +B_{1}t+B_0)e^{\alpha t}\sin \beta t,\end{array}$$

Where $k$ is the larger value of $m$ and $n$

• $s=0$ if $\alpha +\beta i$ is not a root of the associated auxiliary equation
• $s=1$ if $\alpha +\beta i$ is a root of the associated auxiliary equation

Variation of Parameters

$$\mathit{y}(t)=v_1(t)y_1(t)+v_2(t)y_2(t)$$

If $y_1$ and $y_2$ are two linearly independent solutions to the corresponding homogeneous equation, then a particular solution to the nonhomogeneous equation is

$$y(t)={\mathit{v}}_1(t)y_1(t)+{\mathit{v}}_2(t)y_2(t)$$

where $v_1^{\prime }$ and $v_2^{\prime }$ are determined by the equations

$$\begin{array}{rl}& v_1^{\prime }{y}_1+{\mathit{v}}_2^{\prime }{y}_2=0\\ & v_1^{\prime }{y}_1^{\prime }+{\mathit{v}}_2^{\prime }{y}_2^{\prime }=f(t)/a.\end{array}$$

To solve

$$D=\left|\begin{array}{rr}{y}_1& y_2\\ y_1^{\prime }& y_2^{\prime }\end{array}\right|$$

$$D_1=\left|\begin{array}{rr}0& y_2\\ \frac{f(t)}{a}& y_2^{\prime }\end{array}\right|\Rightarrow v_1=\int \frac{D_1}{D}dt$$

$$D_2=\left|\begin{array}{rr}{y}_1& 0\\ y_1^{\prime }& \frac{f(t)}{a}\end{array}\right|\Rightarrow v_2=\int \frac{D_2}{D}dt$$

$$y_p(t)={\mathit{v}}_1(t)y_1(t)+{\mathit{v}}_2(t)y_2(t)$$

Initial Value

Take the n'th derivative, plug in values given for each, use system of equations to solve for constants.
usually the initial conditions will be a function of 0 to cancel things out

$$\begin{array}{r}y(t)=c_1{e}^{r_{1}t}+c_2{e}^{r_{2}t}\\ y^{\prime }(t)=r_1{c}_1{e}^{r_{2}t}+r_2{c}_2{e}^{r_{2}t}\end{array}$$

$$y(0)=n$$

$$y(0)=m$$

$$n=c_1{e}^{r_1(0)}+c_2{e}^{r_2(0)}\Rightarrow n=c_1+c_2$$

$$m=r_1{c}_1{e}^{r_2(0)}+r_2{c}_2{e}^{r_2(0)}\Rightarrow m=r_1{c}_1+r_2{c}_2$$