Post on 27-Jun-2020
September 6 Math 2306 sec. 51 Fall 2019
Section 4: First Order Equations: Linear
General Solution of First Order Linear ODEI Put the equation in standard form y ′ + P(x)y = f (x), and correctly
identify the function P(x).I Obtain the integrating factor µ(x) = exp
(∫P(x)dx
).
I Multiply both sides of the equation (in standard form) by theintegrating factor µ. The left hand side will always collapse intothe derivative of a product
ddx
[µ(x)y ] = µ(x)f (x).
I Integrate both sides, and solve for y .
y(x) =1
µ(x)
∫µ(x)f (x)dx = e−
∫P(x) dx
(∫e∫
P(x) dx f (x)dx + C)
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Solve the ODE
dydx
+y = 3xe−x
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Solve the IVP
xdydx−y = 2x2, x > 0 y(1) = 5
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VerifyJust for giggles, lets verify that our solution y = 2x2 + 3x really doessolve the differential equation we started with
xdydx− y = 2x2.
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Steady and Transient States
For some linear equations, the term yc decays as x (or t) grows. Forexample
dydx
+ y = 3xe−x has solution y =32
x2e−x + Ce−x .
Here, yp =32
x2e−x and yc = Ce−x .
Such a decaying complementary solution is called a transient state.
The corresponding particular solution is called a steady state.
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