Techniques of Applied Mathematics Ordinary Differential Equations And Greens F
Techniques of Applied Mathematics Ordinary Differential Equations And Greens F
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The proof of this theorem will follow from Theorem III-l of Appendix III.
Theorem X . If a. Is a root cf the indicial equation such that a + ra is o o not a root (m is any positive integer), then the differential equation has a solution of the folloxd. Ng form ; "o, 2, u = X (a + a^x + a^x + ... ) .
Problems ; 3. 30. Expand the solutions of the hyper geometric equation x(l-x) u" + f Y - (a+P+1) x3 u' - apu » in povrer series about the origin. 3. 31. Expand the solutions of Legendre's equation [... (l-x^)u'] ' + vu - in power series around the point x = 1. Hint. Put x = 1+t in the differ- n ential equation. Assume u = t (1+a^ t+a„t + ... ), and continue as for Bessel's equation.
- 154 - 3. 32. Consider the equation u» + q-. U' + QpU = where q, (x), q^Cx) are each the quotients of two power series in x. Show that the indicial equation at x = has degree two if and only if xq, (x) 2 and X qp(x) are both bounded as x approaches zero.
3. 33. "Consider the equation u" + q-, u' + q^u = 0. Suppose that in the neighborhood of infinity, q, and q^ can be expanded in power series in 1/x.
What to read after Techniques of Applied Mathematics Ordinary Differential Equations And Greens F?
You can find similar books in the "Read Also" column, or choose other free books by Bernard Friedman to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
The proof of this theorem will follow from Theorem III-l of Appendix III.
Theorem X . If a. Is a root cf the indicial equation such that a + ra is o o not a root (m is any positive integer), then the differential equation has a solution of the folloxd. Ng form ; "o, 2, u = X (a + a^x + a^x + ... ) .
Problems ; 3. 30. Expand the solutions of the hyper geometric equation x(l-x) u" + f Y - (a+P+1) x3 u' - apu » in povrer series about the origin. 3. 31. Expand the solutions of Legendre's equation [... (l-x^)u'] ' + vu - in power series around the point x = 1. Hint. Put x = 1+t in the differ- n ential equation. Assume u = t (1+a^ t+a„t + ... ), and continue as for Bessel's equation.
- 154 - 3. 32. Consider the equation u» + q-. U' + QpU = where q, (x), q^Cx) are each the quotients of two power series in x. Show that the indicial equation at x = has degree two if and only if xq, (x) 2 and X qp(x) are both bounded as x approaches zero.
3. 33. "Consider the equation u" + q-, u' + q^u = 0. Suppose that in the neighborhood of infinity, q, and q^ can be expanded in power series in 1/x.
What to read after Techniques of Applied Mathematics Ordinary Differential Equations And Greens F?
You can find similar books in the "Read Also" column, or choose other free books by Bernard Friedman to read onlineMoreLess
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