MTH232 Tma Solutions
1. In the special case in question (7) where \(x_0=0) this series is also called
Fibonnaci sequence
Taylor series
—>> Maclaurin series
all of the above
2. For the Legendre’s equation \(\large \((1-x^2)y{”}-2xy{‘}+\alpha (\alpha +1)y=0) what type of point is \(x_0=1)\)n
—>> regular singular point
Irregular point
Singular point
Irregular singular point
3. \(\large A set of function \({f_0(x),f_1(x),…..,f_n(x),…})\) is said to be orthogonal wrt w(x) over the interval (a,b) if
—>> \(\large \( \int_{a}^{b}w(x)f_n(x)f_m(x)dx = \left\{ \begin{array}{ll} 1 & \mbox{if $m \neq n$};\\ 0 & \mbox{if $m=n$}.\end{array} \right. \)\)
\(\large \( \int_{a}^{b}w(x)f_n(x)f_m(x)dx = \left\{ \begin{array}{ll} 0 & \mbox{if $m \neq n$};\\ 1 & \mbox{if $m=n$}.\end{array} \right. \)\)
Both A and B
None of the above
4. If a function f has derivatives of all orders at a point \(\large \(x=x_0)\) then the Taylor series of f about \(x_0) is defined byn
\(\large \( \sum_{n=0}^\infty \frac{f^n(x_0)}{n!}\)\)
—>> \(\large \( \sum_{n=0}^\infty \frac{f^n(x_0)}{n!} (x-1)^n)\)\)
\(\large \( \sum_{n=0}^\infty \frac{f^n(x_0)}{n-1}\)\)
None of the above
5. Find the radius of convergence of the series \(\large \( \sum_{n=1}^\infty 2^nn^2(x-1)^2\)n
—>> \(\large \(R=\frac{1}{2})\)
\(\large \(R=\frac{1}{3})\)
\(\large \(R=\frac{3}{5})\)
\(\large \(R=\frac{1}{7})\)
6. If g(x) is an even function then \(\(\int_{-c}^{c}g(x)dx=?)\)n
\(\(\int_{2}^{c}g(x)dx)\)
—>> \(\(2\int_{0}^{c}g(x)dx)\)
\(\(2\int_{c}^{0}g(x)dx)\)
\(\(\int_{-2}^{0}g(x)dx)\)
7. \(Solve \(ay\frac{dy}{dx}+4x=0)\)
\(\(y=\sqrt\frac{2c-4x{2}}{18})\)
\(\(y=\sqrt\frac{2c-4x{2}}{3c})\)
—>> \(\(y=\sqrt\frac{2c-4x{2}}{9})\)
\(\(y=\sqrt\frac{5c-2x{2}}{9})\)
8. \(What is the order of the following DE \(\(\frac{dy}{dx})^2+2y=1\)
Second order
—>> First order
zero order
third order
9. What type of point is \(x_0=0) for the Bessel’s equation \(\large \(x^2y{”}+xy{‘}+(x^2-v^2)y=0)\)n
Singular point
Irregular point
Irregular singular point
—>> regular singular point
10. When is a D.E is said to be separable ?
—>> when it can be put in the form \(\(g(y)dy=f(x)dx)\)n
when it can be put in the form \(\frac{g(y)dy}{f(x)dx})\)n n
Both A and B
None of the above
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