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pshichka [43]
3 years ago
13

See comment for what I need

Mathematics
1 answer:
Roman55 [17]3 years ago
4 0

Answer:

I already sent u the link from there, even tho, they deleted my question, but yh i set the title as well incase u needed it! Good luck

Step-by-step explanation:

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Find a linear second-order differential equation f(x, y, y', y'') = 0 for which y = c1x + c2x3 is a two-parameter family of solu
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Let y=C_1x+C_2x^3=C_1y_1+C_2y_2. Then y_1 and y_2 are two fundamental, linearly independent solution that satisfy

f(x,y_1,{y_1}',{y_1}'')=0
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Note that {y_1}'=1, so that x{y_1}'-y_1=0. Adding y'' doesn't change this, since {y_1}''=0.

So if we suppose

f(x,y,y',y'')=y''+xy'-y=0

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To make sure everything cancels out, multiply the second degree term by -\dfrac{x^2}3, so that

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as desired. So one possible ODE would be

-\dfrac{x^2}3y''+xy'-y=0\iff x^2y''-3xy'+3y=0

(See "Euler-Cauchy equation" for more info)
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