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Method of undermined coefficients:

Solve: $\\displaystyle \\frac{d^2y}{dx^2}+2a\\frac{dy}{dx}+a^2y=0,\\;y(0)=c$ and $y(1)=d$. (Equal roots example). Includes an interactive plot.

rebelmaths

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{graphdisplay()}

Solution is:

$y=\\;\\;$[[0]]

Input all numbers correct to 3 decimal places.

Your solution is graphed above once you have input the solution before submitting.

Note that the points $(0,\\var{c}),\\;(1,\\var{d})$ which the curve has to go through are marked on the graph.

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Method of undetermined coefficients:

Find the solution of:
\\[\\simplify[std]{(d^2y/dx^2)+{2*a}*(dy/dx)+{a^2}y}=0\\]
which satisfies $y(0)=\\var{c}$ and $y(1)=\\var{d}$.

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The auxillary equation is $\\simplify[std]{lambda^2+{2*a}lambda+{a^2}}=0$.

On solving this equation we get $\\lambda=\\var{-a}$ twice.

Hence the general solution is:
\\[y = \\simplify[std]{A*e^({-a}x)+B*x*e^({-a}x)}\\]
The boundary conditions give:

$y(0)=\\var{c} \\Rightarrow A=\\var{c}$

$y(1)=\\var{d} \\Rightarrow \\simplify[std]{Ae^{-a}+Be^{-a}={d}}\\Rightarrow A+B = \\simplify[std1]{{d}e^{a}}$

So $B=\\simplify[std1]{{d}e^{a}-{c}}=\\var{f1}$ to 3 decimal places.

Hence the solution is:
\\[y=\\simplify[std1]{(({c} * Exp(({( - a)} * x))) + ({f1} * x * Exp(({( - a)} * x))))}\\]

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