# MTH232 Solutions

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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