Find ODE having the given function as its general solution.y=x2+c1e2x+c2e3x

Name: Find ODE having the given function as its general solution.y=x2+c1e2x+c2e3x
Uploaded: 2022-02-23T17:22:57+00:00
Description: Find ODE having the given function as its general solution.y=x2+c1e2x+c2e3x

Question

Leonardo Mcpherson · Accepted Answer

To find the ordinary differential equation of the given function as its general solution

y = x^{2} + c_{1} e^{2 x} + c_{2} e^{3 x}

Solution:

y = x^{2} + c_{1} e^{2 x} + c_{2} e^{3 x}

as we can see

x^{2}

is the particular solution of the equation denoted by

y_{p} (x) = x^{2}

As we can determine from the complementary solution that 2 and 3 must be the solution of the equation so, we can write the characteristic equation as

(r - 2) (r - 3) = r^{2} - 5 r + 6

So its associated homogeneous equation is of the form

y^{″} - 5 y^{'} + 6 y = 0

We will find the non homogeneous ordinary differential equation that is of the form

y^{″} - 5 y^{'} + 6 y = p x^{2} + q x + r

(1)
As the general second order particular solution is of the form

y_{p} (x) = A x^{2} + B x + C

(2)

y_{p}^{'} (x) = 2 A X + B

(3)

y {^{″}}_{p} (x) = 2 A

(4)
Substituting (2),(3),(4) in (1)

2 A - 5 (2 A x + B) + 6 (A x^{2} + B x + C) = p x^{2} + q x + r

(5)
Now we are give

y_{p} (x) = x^{2}

On equating (1) and

y_{p} (x) = x^{2}

we get:

A = 1, B = 0, C = 0

Putting

A = 1, B = 0, C = 0

in equation (5) we get:

2 - 10 x + 6 x^{2} = p x^{2} + q x + r

On equating the coefficients we get:

p = 6, q = - 10, r = 2

This implies the non homogeneous equation is

y^{″} - 5 y + 6 e = 6 x^{2} - 10 x + 2

So, the ODE of the general solution

y = x^{2} + c_{1} e^{2 x} + c_{2} e^{3 x}

is

y^{″} - 5 y^{'} + 6 y = 6 x^{2} - 10 x + 2

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Find ODE having the given function as its

Answered question

Answer & Explanation

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