Key Points of MA 1506

Recall Key Points of MA 1505:

1. Fundamental Theorem in Calculus:

(\int_{0}^{x} f(t)dt)^{'}=f(x)

2. Integration by Parts:

\int f(x) dg(x)= f(x)g(x) - \int g(x) df(x)

3. Derivatives and Integration:

( \sin x)^{'} = \cos x

(\cos x)^{'} = -\sin x

Hyperbolic Sine: \sinh x= \frac{e^{x}- e^{-x}}{2}

Hyperbolic Cosine: \cosh x=\frac{e^{x}+e^{-x}}{2}

(\sinh x)^{'}= \cosh x

(\cosh x)^{'}= \sinh x

MA 1506 Tutorials:

Ordinary Differential Equations:

1. Newton-Leibniz Formula

y=y(x) is a function with one variable x with ordinary differential equation y^{'}=f(x). The solution is

y=\int f(x) dx + C with some constant C .

2. Separable Equations

y=y(x) is a function with one variable x with ordinary differential equation N(y)y^{'}=M(x), where N(y) is a function with one variable y and M(x) is a function with one variable x.

The solution is \int M(x) dx = \int N(y) dy + C with some constant C.

3. One Order Ordinary Differential Equations

y=y(x) is a function with one variable x with one order ordinary differential equation y^{'}+P(x)y=Q(x). The integrating factor is R(x)= exp ( \int P(x) dx). That means

d( R(x) y) = R(x) Q(x) and take the integration under x at the both sides,

R(x)y=\int R(x)Q(x) dx + C for some constant C.

Sometimes, we need to make some substitution as z=y^{2} or z=\frac{1}{y} , since the following formulas:

2 y y^{'} = (y^{2})^{'},

-\frac{y^{'}}{y^{2}} = (\frac{1}{y})^{'}.

If there is an initial condition y(0)=A for the first order ordinary differential equation y^{'} + P(x)y=Q(x), then we must make use of the initial condition to calculate the constant C after we solved the equation.

4. Second Order Ordinary Differential Equations

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