Solve a constant coefficient second order ODE,

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Bill Foster 6 years, 3 months ago

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

The general solution of the differential equation

$\simplify{{a1}*y''-{b1}*y}=0$

can be written in the form

$y(x)=A\mathrm{e}^{\lambda_1 x}+B\mathrm{e}^{\lambda_2 x},$

where $A,B,\lambda_1,\lambda_2\in\mathbb{R}$ and $\lambda_1>\lambda_2$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	number	4.028
b	number	-3.028
a1	integer	7
b1	integer	9
tol	number	0.001
c1	number	1
d1	number	8

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Value

xxxxxxxxxx
 
precround(1/2*(c1+d1*(sqrt(a1/b1))),3)

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Generated value: `number`

4.028

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1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Part b) Gap-fill
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Solve the differential equation, and enter the values of $\lambda_1$ and $\lambda_2$ in the boxes.

Do not enter decimals in your answers. If you need to enter a square root, e.g. $\sqrt{x}$ , enter this as sqrt(x).

$\lambda_1=$ Gap 0.

$\lambda_2=$ Gap 1.

Advice

a)

We can solve the differential equation by making the assumption that $y=\mathrm{e}^{\lambda x}$ .

By substituting this expression for $y$ into the equation, and cancelling terms in $\mathrm{e}^{\lambda x}$ (which we can do, because $\mathrm{e}^{\lambda x}\ne 0$ ), we obtain

$\simplify{{a1}*lambda^2-{b1}}=0,$

which has solutions

$\lambda_{1,2}=\pm\simplify{sqrt({b1}/{a1})}.$

The general solution to the differential equation, therefore, is

$y(x)=A\exp\left(\simplify{sqrt({b1})/sqrt({a1})*x}\right)+B\exp\left(\simplify{-sqrt({b1})/sqrt({a1})*x}\right).$

b)

Applying the conditions $y(0)=\var{c1}$ and $y'(0)=\var{d1}$ gives

$\var{c1}=y(0)=A+B$

and

$\var{d1}=y'(0)=\simplify{A*(sqrt({b1})/sqrt({a1}))-B*(sqrt({b1})/sqrt({a1}))}.$

Some rearrangement then gives

$A=\simplify{1/2*({c1}+{d1}*(sqrt({a1})/sqrt({b1})))}=\var{A}\;\text{to 3 d.p.,}$

and

$B=\simplify{1/2*({c1}-{d1}*(sqrt({a1})/sqrt({b1})))}=\var{B}\;\text{to 3 d.p.}$

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Solve a constant coefficient second order ODE,

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Bill Foster 6 years, 3 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Solve a constant coefficient second order ODE,

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

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Bill Foster 6 years, 3 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: number

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a)

b)

Generated value: `number`