Mixing
Power Series and Recursion
Clash Royale CLAN TAG#URR8PPP
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Consider the equation: $$(1-x^2)y'' -xy' +alpha^2 y=0$$ where $alpha$ is a real number.
For $|x|<1$ and all values of $alpha$, look for a fundamental set of solutions $y_1$, $y_2$ which are power series in $x$. Prove that they are linearly independent. You can leave the formula for the coefficients in recursive form, but compute the first three terms of $y_1$, $y_2$ explicitly in terms of $alpha$.
What does it mean when the question asserts to look for a fundamental set of solutions that are a power series in $x$? Likewise, what does the question mean when it states to leave the coefficients in recursive form?
differential-equations power-series recursion independence
edited yesterday
Rócherz
2,1811417
asked yesterday
seekingalpha23
156
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Consider the equation: $$(1-x^2)y'' -xy' +alpha^2 y=0$$ where $alpha$ is a real number.
For $|x|<1$ and all values of $alpha$, look for a fundamental set of solutions $y_1$, $y_2$ which are power series in $x$. Prove that they are linearly independent. You can leave the formula for the coefficients in recursive form, but compute the first three terms of $y_1$, $y_2$ explicitly in terms of $alpha$.
What does it mean when the question asserts to look for a fundamental set of solutions that are a power series in $x$? Likewise, what does the question mean when it states to leave the coefficients in recursive form?
differential-equations power-series recursion independence
edited yesterday
Rócherz
2,1811417
asked yesterday
seekingalpha23
156
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider the equation: $$(1-x^2)y'' -xy' +alpha^2 y=0$$ where $alpha$ is a real number.
For $|x|<1$ and all values of $alpha$, look for a fundamental set of solutions $y_1$, $y_2$ which are power series in $x$. Prove that they are linearly independent. You can leave the formula for the coefficients in recursive form, but compute the first three terms of $y_1$, $y_2$ explicitly in terms of $alpha$.
What does it mean when the question asserts to look for a fundamental set of solutions that are a power series in $x$? Likewise, what does the question mean when it states to leave the coefficients in recursive form?
differential-equations power-series recursion independence
edited yesterday
Rócherz
2,1811417
asked yesterday
seekingalpha23
156
Consider the equation: $$(1-x^2)y'' -xy' +alpha^2 y=0$$ where $alpha$ is a real number.
For $|x|<1$ and all values of $alpha$, look for a fundamental set of solutions $y_1$, $y_2$ which are power series in $x$. Prove that they are linearly independent. You can leave the formula for the coefficients in recursive form, but compute the first three terms of $y_1$, $y_2$ explicitly in terms of $alpha$.
What does it mean when the question asserts to look for a fundamental set of solutions that are a power series in $x$? Likewise, what does the question mean when it states to leave the coefficients in recursive form?
differential-equations power-series recursion independence
edited yesterday
Rócherz
2,1811417
asked yesterday
seekingalpha23
156
edited yesterday
Rócherz
2,1811417
edited yesterday
Rócherz
2,1811417
edited yesterday
Rócherz
2,1811417
2,1811417
asked yesterday
seekingalpha23
156
asked yesterday
seekingalpha23
156
asked yesterday
seekingalpha23
156
156
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
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0
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It means find a solution of the form
$$
y(x)=sum_n=0^infty a_nx^n,
$$
were you are supposed to find the coefficients $a_n$. To leave them in recursive form means that you do not need to find them explicitly, but to obtain a recurrence formula where $a_n$ is given in terms of $a_0,dots,a_n-1$.
answered yesterday
Julián Aguirre
64.3k23894
Proving linear independence is easy for me. I will simply take the Wronskian and show that it does not equal to zero. But how would I go about finding y1,y2,y3. Shouldn't there at most be only two solutions? I'm lost at making my first step
– seekingalpha23
20 hours ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
It means find a solution of the form
$$
y(x)=sum_n=0^infty a_nx^n,
$$
were you are supposed to find the coefficients $a_n$. To leave them in recursive form means that you do not need to find them explicitly, but to obtain a recurrence formula where $a_n$ is given in terms of $a_0,dots,a_n-1$.
answered yesterday
Julián Aguirre
64.3k23894
Proving linear independence is easy for me. I will simply take the Wronskian and show that it does not equal to zero. But how would I go about finding y1,y2,y3. Shouldn't there at most be only two solutions? I'm lost at making my first step
– seekingalpha23
20 hours ago
add a comment |Â
up vote
0
down vote
It means find a solution of the form
$$
y(x)=sum_n=0^infty a_nx^n,
$$
were you are supposed to find the coefficients $a_n$. To leave them in recursive form means that you do not need to find them explicitly, but to obtain a recurrence formula where $a_n$ is given in terms of $a_0,dots,a_n-1$.
answered yesterday
Julián Aguirre
64.3k23894
Proving linear independence is easy for me. I will simply take the Wronskian and show that it does not equal to zero. But how would I go about finding y1,y2,y3. Shouldn't there at most be only two solutions? I'm lost at making my first step
– seekingalpha23
20 hours ago
add a comment |Â
up vote
0
down vote
up vote
0
down vote
It means find a solution of the form
$$
y(x)=sum_n=0^infty a_nx^n,
$$
were you are supposed to find the coefficients $a_n$. To leave them in recursive form means that you do not need to find them explicitly, but to obtain a recurrence formula where $a_n$ is given in terms of $a_0,dots,a_n-1$.
answered yesterday
Julián Aguirre
64.3k23894
It means find a solution of the form
$$
y(x)=sum_n=0^infty a_nx^n,
$$
were you are supposed to find the coefficients $a_n$. To leave them in recursive form means that you do not need to find them explicitly, but to obtain a recurrence formula where $a_n$ is given in terms of $a_0,dots,a_n-1$.
answered yesterday
Julián Aguirre
64.3k23894
answered yesterday
Julián Aguirre
64.3k23894
answered yesterday
Julián Aguirre
64.3k23894
answered yesterday
Julián Aguirre
64.3k23894
64.3k23894
Proving linear independence is easy for me. I will simply take the Wronskian and show that it does not equal to zero. But how would I go about finding y1,y2,y3. Shouldn't there at most be only two solutions? I'm lost at making my first step
– seekingalpha23
20 hours ago
add a comment |Â
Proving linear independence is easy for me. I will simply take the Wronskian and show that it does not equal to zero. But how would I go about finding y1,y2,y3. Shouldn't there at most be only two solutions? I'm lost at making my first step
– seekingalpha23
20 hours ago
Proving linear independence is easy for me. I will simply take the Wronskian and show that it does not equal to zero. But how would I go about finding y1,y2,y3. Shouldn't there at most be only two solutions? I'm lost at making my first step
– seekingalpha23
20 hours ago
Proving linear independence is easy for me. I will simply take the Wronskian and show that it does not equal to zero. But how would I go about finding y1,y2,y3. Shouldn't there at most be only two solutions? I'm lost at making my first step
– seekingalpha23
20 hours ago
add a comment |Â
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