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Second plus order ode formulas reference

Homogeneous Equations

Characteristic Equations

\[\begin{align*} ay^{\prime \prime}+by^{\prime}+cy = f(t) && \text{OR} && a \frac{d^{2}y}{dt^{2}}+b \frac{dy}{dt}+cy = f(t) \\ \\ && f(t) = 0 \end{align*}\]
\[y = e^{rt}\]

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.

\[ ar^{2}e^{rt}+bre^{rt}+ce^{rt} = 0\]
\[\Rightarrow\]
\[e^{rt}(ar^{2}+br+c) = 0\]

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

\[\begin{align*}ar^2+br+c=0\end{align*}\]

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

\[\begin{align*} r_{1} = \frac{-b+\sqrt{b^{2}-4ac}}{2a} && \text{And} && r_{2} = \frac{-b-\sqrt{b^{2}-4ac}}{2a}\end{align*}\]

General Solutions to Characteristic Equations

  • Quadratics
  • 2 Real, Unique Roots
\[\begin{align*} y(t) = c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t}\end{align*}\]
  • double root
\[\begin{align*} y(t) = c_{1}e^{rt}+c_{2}e^{rt}t \end{align*}\]
  • 2 Complex Roots
\[r = \alpha \pm \beta i\]
\[\begin{align*} y(t) = c_{1}e^{\alpha t}\cos(\beta t) + c_{2}e^{\alpha t}\sin(\beta t) \end{align*}\]
  • Cubic
  • 3 Real, Unique Roots
\[\begin{align*} y(t) = c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t}+c_{3}e^{r_{3}t} \\ \end{align*}\]
  • 3 Real Roots, 1 Double Root
\[\begin{align*} y(t) = c_{1}e^{r_{1}t}+c_{2}e^{rt}+c_{3}e^{rt}t \\ \end{align*}\]
  • 1 Real Root, 2 Complex Roots
\[\begin{align*} y(t) = c_{1}e^{rt}+c_{2}e^{\alpha t}\cos(\beta t) + c_{3}e^{\alpha t}\sin(\beta t) \\ \end{align*}\]
Cauchy-Euler Equations
\[\begin{align*} at^{2}y^{\prime\prime}+bty^{\prime}+cy = 0 \\ \\ \text{OR} \quad\quad\quad\quad \\ \\ y^{\prime\prime}+ \frac{b}{at}y^{\prime}+ \frac{c}{at^{2}} = 0 \end{align*}\]

assuming that \(t>0\)

\[\begin{alignat*}{2} &y(t) = t^{r} \\ \\&\frac{dy}{dt} = rt^{r-1} \\ \\ &\frac{d^{2}y}{dt^{2}} = (r^{2}-r) t^{r-2} \\ \\ &\text{plugging into the original formula} \\ \end{alignat*}\]
\[\begin{align*} at^{2} (r^{2}-r)t^{r-2}+bt(r)t^{r-1} +ct^{r} = 0 \\ \\ \end{align*}\]
\[\begin{align*} \frac{at^{2}(r^{2}-r)t^{r}}{t^{2}} + \frac{bt(r)t^{r}}{t} + ct^{r}=0$a(r^{2}-r)t^{r}+ b(r)t^{r}+ct^{r}=0 \end{align*}\]
\[t^{r}(ar^{2}-ar +br + c)=0 \quad \Rightarrow \quad ar^{2}-ar +br + c=0\]
\[ar^{2}+(b-a)r + c = 0\]

General Solutions to Cauchy-Euler Equations

Real, unique roots

\[y(t) = c_{1} t^{r_{1}}+ c_{2}t^{r_{2}}\]

Double root

\[y(t) = c_{1}t^{r}+c_{2}t^{r} \ln(t)\]

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

\[\begin{align*} ay^{\prime \prime} + by^{\prime} + cy = f(t) && \text{OR} && a \frac{d^{2}y}{dt^{2}}+b \frac{dy}{dt}+cy = f(t) \end{align*}\]
\[f(t) \ne 0\]

Homogeneous Solution

\[ay^{\prime \prime} + by^{\prime} + cy = 0 \quad \Rightarrow \quad y_c(t)\]

Non Homogeneous Solution

\[ay^{\prime \prime} + by^{\prime} + cy = f(t) \quad \Rightarrow \quad y_p(t)\]

General Solution

\[y(t) = y_{p}(t)+ y_{c}(t)\]

Linear Equations (Constant Coefficients)

Undetermined Coefficients
\[a y^{\prime \prime}+b y^{\prime}+c y=C t^{m} e^{r t}\]

where \(m\) is a nonnegative integer, use the form

\[y_{p}(t)=t^{s}\left(A_{m} t^{m}+\cdots+A_{1} t+A_{0}\right) e^{r t}\]
  • \(s=0\) 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
\[a y^{\prime \prime}+b y^{\prime}+c y=\left\{\begin{array}{c} C t^{m} e^{\alpha t} \cos \beta t \\ \text { or } \\ C t^{m} e^{\alpha t} \sin \beta t \end{array}\right.\]

for \(\beta \neq 0\), use the form

\[\begin{aligned} y_{p}(t)=& t^{s}\left(A_{m} t^{m}+\cdots+A_{1} t+A_{0}\right) e^{\alpha t} \cos \beta t \\ &+t^{s}\left(B_{m} t^{m}+\cdots+B_{1} t+B_{0}\right) e^{\alpha t} \sin \beta t, \end{aligned}\]
  • \(s=0\) 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.
\[a y^{\prime \prime}+b y^{\prime}+c y=P_{m}(t) e^{r t},\]

where \(P_{m}(t)\) is a polynomial of degree \(m\), use the form

\[y_{p}(t)=t^{s}\left(A_{m} t^{m}+\cdots+A_{1} t+A_{0}\right) e^{r t} ;\]
  • \(s=0\) 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
\[a y^{\prime \prime}+b y^{\prime}+c y=P_{m}(t) e^{\alpha t} \cos \beta t+Q_{n}(t) e^{\alpha t} \sin \beta t, \quad \beta \neq 0,\]

where \(P_{m}(t)\) is a polynomial of degree \(m\) and \(Q_{n}(t)\) is a polynomial of degree \(n\), use the form

\[\begin{aligned} y_{p}(t)=& t^{s}\left(A_{k} t^{k}+\cdots+A_{1} t+A_{0}\right) e^{\alpha t} \cos \beta t \\ &+t^{s}\left(B_{k} t^{k}+\cdots+B_{1} t+B_{0}\right) e^{\alpha t} \sin \beta t, \end{aligned}\]

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
\[\boldsymbol{y}(t)=v_{1}(t) y_{1}(t)+v_{2}(t) y_{2}(t)\]

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

\[y(t)=\boldsymbol{v}_{1}(t) y_{1}(t)+\boldsymbol{v}_{2}(t) y_{2}(t)\]

where \(v_{1}^{\prime}\) and \(v_{2}^{\prime}\) are determined by the equations

\[\begin{aligned} &v_{1}^{\prime} y_{1}+\boldsymbol{v}_{2}^{\prime} y_{2}=0 \\ &v_{1}^{\prime} y_{1}^{\prime}+\boldsymbol{v}_{2}^{\prime} y_{2}^{\prime}=f(t) / a . \end{aligned}\]

To solve

\[D =\left | {\begin{array}{rr}y_1&y_2\\y_1^{\prime}&y_2^{\prime}\end{array}} \right | \]
\[D_1 =\left | {\begin{array}{rr}0&y_2\\\frac{f(t)}{a}&y_2^{\prime}\end{array}} \right | \quad \Rightarrow \quad v_{1}= \int \frac{D_{1}}{D} dt\]
\[D_2 =\left | {\begin{array}{rr}y_1&0\\y_{1}^{\prime}&\frac{f(t)}{a}\end{array}} \right | \quad \Rightarrow \quad v_{2}= \int \frac{D_{2}}{D} dt\]
\[y_p(t)=\boldsymbol{v}_{1}(t) y_{1}(t)+\boldsymbol{v}_{2}(t) y_{2}(t)\]

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

\[\begin{align*} y(t) = c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t} \\ y^{\prime}(t)= r_{1}c_{1}e^{r_{2}t}+r_{2}c_{2}e^{r_{2}t}\end{align*}\]
\[y(0) = n\]
\[y(0) = m\]
\[n=c_{1}e^{r_{1}(0)}+c_{2}e^{r_{2}(0)} \quad\Rightarrow\quad n = c_{1}+c_{2} \]
\[m= r_{1}c_{1}e^{r_{2}(0)}+r_{2}c_{2}e^{r_{2}(0)} \quad \Rightarrow \quad m=r_{1}c_{1}+r_{2}c_2\]