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Double real root. $\Delta=0$. $\displaystyle m=-\frac {b-a} {2a}$. $y_h= [C_1+C_2\ln|x|]x^ {m}$. 1. Diﬀerentiate both sides of the equation with respect to “x” 2. When taking the derivative of any term that has a “y” in it multiply the term by y0 (or dy=dx) 3.

ODE Cheat Sheet First Order Equations Separable Ry0(x) = f(x)g(y) dy g(y) = R f(x)dx+C Linear First Order y0(x)+p(x)y(x) = f(x) (x) = exp R x p(˘)d˘ Integrating factor. ( y)0= f Exact Derivative. Solution: y(x) = 1 (x) nR f(˘) (˘)d˘+ C Exact 0 = M(x;y)dx+ N(x;y)dy Solution: u(x;y) = const where du= @u @x dx+ @u @y dy @u @x = M(x;y); @u @y = N(x;y) Condition: My= Nx

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Fischer equation, nonlinear PDE for modeling population growth. $u_t-Du_{xx}=r u(x,t) (1-\frac{u(x,t)}{K})$ where $K$ is carrying capacity, and $r$ is growth rate. 2.3 Hyperbolic Some properties A differential equation is an equation for a function containing derivatives of that function. For exam-ple, the differential equations for an RLC circuit, a pendulum, and a diffusing dye are given by L d2q dt2 + R dq dt + 1 C q = E 0 coswt, (RLC circuit equation) ml d2q dt2 +cl dq dt ODE Cheat Sheet First Order Equations Separable Ry0(x) = f(x)g(y) dy g(y) = R f(x)dx+C Linear First Order y0(x)+p(x)y(x) = f(x) (x) = exp R x p(˘)d˘ Integrating factor. ( y)0= f Exact Derivative. Solution: y(x) = 1 (x) nR f(˘) (˘)d˘+ C Exact 0 = M(x;y)dx+ N(x;y)dy Solution: u(x;y) = const where du= @u @x dx+ @u @y dy @u @x = M(x;y); @u @y = N(x;y) Condition: My= Nx 2014-03-07 AP Calculus: Differential Equations Cheat Sheet Requirements.
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Here is what I would consider writing if I were a student in 2014-03-07 · Di erential Equations Study Guide1 First Order Equations General Form of ODE: dy dx = f(x;y(1) ) (2) Initial Value Problem: y0= f(x;y); y(x 0) = y 0 Linear Equations General Form: y0+p(x)y= f(x(3) ) Integrating Factor: (x) = e R (4) p(x)dx =) d dx ( (x)y) = (x)f(x(5) ) General Solution: y= 1 (x) Z (x)f(x)dx+C (6) Homogeneous Equations General Form: y0= f(y=x(7) ) AP Calculus: Differential Equations Cheat Sheet Requirements. Name: _____ To receive credit: Cheat sheets may be written on 8.5” by 11” paper; including notebook paper, printer paper, and ORDINARY DIFFERENTIAL EQUATION REVIEW SHEET I [28/10/2015] IMPORTANT EQUATIONS R R. Explicit Dependent variable can be separated, x + 2y = 0. udv = uv vdu d(lnx) = x1 dx d (sin x) = cos(x) dx d (cos x) = sin(x) dx d (tan x) = sec2 (x) dx d (csc x) = csc(x) cot(x)x dx d (sec x) = sec(x) tan(x) dx d (cot x) = csc2 (x) dx 0. g ( fg )0 = f gf g2 Leibniz Rule: Singular Solution: ??? If you want to learn differential equations, have a look at Differential Equations for Engineers If your interests are matrices and elementary linear algebra, try Matrix Algebra for Engineers If you want to learn vector calculus (also known as multivariable calculus, or calcu-lus three), you can sign up for Vector Calculus for Engineers algebraic subsidiary equation.

Differential Equation Formula Sheet Differential Equation Formula Sheet A differential equation is a part of mathematical equation. The differential equation is used for calculating the unknown Writing Equations Given Slope and a Point If you are given slope and a point, then you are given m, x, and y for the equation y = mx + b.
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Diﬀerentiate both sides of the equation with respect to “x” 2. When taking the derivative of any term that has a “y” in it multiply the term by y0 (or dy=dx) 3. Solve for y0 When ﬁnding the second derivative y00, remember to replace any y0 terms in your ﬁnal answer with the equation for y 0you already found. In other words - Ordinary Differential Equations (ODE) are differential equations having one independent variable. - Partial Differential Equations (PDE) are differential equations having two or more independent variables. 4 references. Elements of partial diﬀerential Equations, Pavel Drabek and Gabriela Holubova, 2007.