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First order non linear differential equations  

Second Order Differential equations

30th July

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First order non-liner differential equations

y=y(x)

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Solve the differential equation and determine the

interval of validity

Example 1

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Solve the differential equation and determine the

interval of validity

Example 1

Put given initial value

c=28

-

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Three possible intervals

Find where initial value y(1)=1/25 lies

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Initial value lies in this interval

Solution

Valid range

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Initial value lies in this interval

Valid range

Behavior of the solution

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Problem 1

Use the initial condition to check the correct sign

Explicit solution for our differential equation

Separate the variable and integrate both sides

To avoid complex number

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Problem 2

Explicit solution

Problem 3

Explicit solution

In(x)x

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Solve the following IVP

Explicit solution.

Problem 4

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Linear Second Order Differential equations

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General form of Linear Second Order Differential equation

When the coefficients are constant

Homogeneous : g(t)=0

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Homogeneous Linear Second Order Differential equation

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Some other solutions

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Linear superposition principle

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y(0)=2, y’(0)=-1

Solve

for the initial conditions

Put initial conditions and solve for the value of the constants

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General steps for solutions:

Assume a solution

Superposition principle

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Find the general solution

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1.

Home work

2.

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Hooke’s law

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Assume a solution

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Wave equation

Initial conditions

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Fourier transform in x using alpha as transform variable

ODE

PDE

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