Question

Cauchy - Euler differential equation!!

(x^2)y" + xy' +4y = cos(2 ln(x)) what is the Cauchy - Euler differential equation general solve?

Answer #1

x^2y'' − 3xy'+ 4y = 0 ; y(1)=5 y'(1)=3
differential equation using the Cauchy-Euler method

x^2y'' − 3xy'+ 4y = 0 ; y(1)=5 y'(1)=3
differential equation using the Cauchy-Euler method

[Cauchy-Euler equations] For the following equations with the
unknown function y = y(x), find the general solution by changing
the independent variable x to et and re-writing the equation with
the new unknown function v(t) = y(et).
x2y′′ +xy′ +y=0
x2y′′ +xy′ +4y=0
x2y′′ +xy′ −4y=0
x2y′′ −4xy′ −6y=0
x2y′′ +5xy′ +4y=0.

Solve the IVP with Cauchy-Euler ODE: x^2 y''+3xy'+4y=0; y(1)=0,
y’(1)=−2

Solve the differential equation by using integrating factors.
xy' = 4y − 6x^2
y(x)=?

Solve the IVP with Cauchy-Euler ODE: x^2 y''+ xy'−16y = 0; y(1)
= 4, y'(1) = 0

Use the substitution x = et to transform the given Cauchy-Euler
equation to a differential equation with constant coefficients.
(Use yp for dy /dt and ypp for d2y/dt2 .) x2y'' + 10xy' + 8y =
x2
Solve the original equation by solving the new equation using
the procedures in Sections 4.3-4.5. y(x) =

Solve the following differential equation using taylor series
centered at x=0:
(2+x^2)y''-xy'+4y = 0

Find the general solution to the Cauchy-Euler equation:
x^2 y'' - 5xy' + 8y = 0

solve differential equation ((x)2 - xy +(y)2)dx - xydy
= 0
solve differential equation (x^2-xy+y^2)dx - xydy =
0

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