Differential Equations

# Differential Equations - is an eqn of the form M(x y dx N(x...

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Differential Equations: Separation of Variables Section 5.7

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y = Ce -2x : general solution (0, 2) : initial condition Solve for C . y = 2 e -2x : particular solution
Separation of Variables Equations are separable if all x terms can be collected with dx and all y terms with d y , and a solution can be obtained by integration. 0 3 2 = + dx dy y x dx x ydy 2 3 - =

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Homogeneous Differential Equations Some differential equations that are not separable in x and y can be made separable by a change of variables. This is true for differential equations of the form y’ = f(x, y) where f is a homogenous function .
The function given by f(x, y) is homogeneous of degree n if f (tx, ty) = t n f(x, y) where n is a real number.

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Defn. of Homogeneous Differential Equation A homogeneous differential eqn.

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Unformatted text preview: is an eqn. of the form M(x, y) dx + N(x, y)dy = 0 Where M and N are homogeneous functions of the same degree. Change of Variables for Homogenous Equations If M (x, y) dx + N(x, y)dy = 0 is homogeneous, then it can be changed into a separable differentia equation by the substitution y = vx where v is a differentable function of x. Orthogonal Trajectories 2 families of curves are mutually orthogonal and each curve in one of the families is called an orthogonal trajectory of the other family of curves if they intersect at right angles. Look at Figure 5.35 pg. 373. Joke Time What is Beethoven doing in his grave? De-composing! Why does a lobster never share? Because it’s shellfish!...
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Differential Equations - is an eqn of the form M(x y dx N(x...

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