You CAN Ace Differential Equations  

17calculus > differential equations > polynomial coefficients  


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Differential Equations Alpha List
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Differential Equations With Polynomial Coefficients  

classify  
secondorder  linear 
homogeneous  polynomial coefficients 
\( r(t)y'' +p(t)y' + q(t)y = 0 \) 
On this page, we discuss differential equations with polynomial coefficients of the form \(\displaystyle{ r(t^n)y^{(n)}(t) + a_{n1}t^{n1}y^{n1}(t) + \cdots + a_0y(t) = g(t) }\), where \(r(t^n)\) is an nthorder polynomial.
For now, we will stick with these two special types of secondorder homogeneous equations.
\(\displaystyle{ t^2y'' +aty' + by = 0 }\)  
\(\displaystyle{ (1t^2)y''  ty' + ay = 0 }\) 
CauchyEuler Differential Equation 
A CauchyEuler Differential Equation (also called EulerCauchy Equation or just Euler Equation) is an equation with polynomial coefficients of the form \(\displaystyle{ t^2y'' +aty' + by = 0 }\).
These may seem kind of specialized, and they are, but equations of this form show up so often that special techniques for solving them have been developed. We will look at a couple of techniques on this page and direct you to other techniques on other pages.
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reduction of order  
Technique 1   Substitution 
A good substitution for this type of equation is \(x=\ln(t)\). This substitution converts the differential equation into one with constant coefficients.
Technique 2   Trial Solution 
A second technique involves using a trial solution \(y=t^k\) where k is a constant.
This video takes you through a general equation and then through 3 examples.  
Chebyshev's Differential Equations 
Chebyshev's equations are of the form \(\displaystyle{ (1t^2)y''  ty' + ay = 0 }\) where a is a real constant. Again, these seem specialized but they occur so often, they are worth discussing separately. These equations can be solved by using the substitution \( t = \cos( \theta )\). This changes the differential equation into a form that can be solved more easily. See the practice problems for examples.
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Practice Problems 

Instructions   Unless otherwise instructed, solve these differential equations using the techniques on this page.