Unit3 & 5
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UNIT‐III Ques ‐1 Find the Laplace Transform of :‐ a) f (t) = Cos t , 0π b) f (t) = 1 , 0
²
, find L(cos²at)
Ques‐3 If f(t) =
, find the Laplace transform of f(t).
Ques – 4 Given L 2
=
/
, Show that L (
√
) = √
Ques‐5 Find the Laplace Transform of t e‐t Sin2t Ques‐6 Using Laplace Transform , evaluate the following integrals:‐ √
(i) dt (ii) dt Ques‐7 Express the Following function in terms of Unit Step Function and find its Laplace Transform:‐ F(t) 1 1 2 t Ques‐8 Find the Laplace Transform of:‐ a) (t‐1)² u (t‐1) b) Sint u (t‐π) Ques‐9 Draw the graph and find the Laplace Transform of the triangular Wave function of Period 2c is given by:‐ F(t) = t ,0
F(t) = 1 , 0
‐1 , Ques‐11 Find the Laplace Transform of the periodic function :‐ F(t)= , for 0 ; F(t+T) = F(t) Ques‐12 (i) Find the inverse Laplace Transform of b) F(p) = a) F(p) = ²
Ques‐13 Use Convolution Theorem ,to evaluate ,
(i) L‐1 {
(ii) L‐1 {
Ques‐14 (i) Solve the Simultaneous Equation by Laplace Transform :‐ , + x = Sint ; Given x(0)=1 , y(0)=0 (ii) Solve the Simultaneous Equation by Laplace Transform :‐ ( D2 – 3) x – 4y =0 x + (D²+1)y =0 For t>0 , given that x=y= 0 = 2 at t = 0
²
UNIT‐V Ques‐1 Use the method of separation of variables , to solve the Equation:‐ 2
(i)
(ii)
² ²
+ u ,given that u(x,0)= 6 2
+ = 0
Ques‐2 A String is stretched and fastened to two points ‘l’ apart. Motion is started by displacing the string in the form, Y= A Sin , from which it is released at time t=0 .Show that the displacement of any point at a distance 'x’ from one end at time ‘t’ is given by, Y(x,t) = A Sin Cos Ques‐3 Show that how the Equation ²
²
c² = ² ² can be solved by the method of separation of variables. If the initial displacement and velocity of a string stretched between x=0 and x=l given by :‐ y=f(x) and = g(x) , determine the constants in the series solution. Ques‐4 Find the temperature in a bar of length 2 whose ends are kept at zero and lateral surface insulated if the initial temperature is given as:‐ Sin + 3 Sin Ques‐5 Use the separation of variables method to solve the equation:‐
² ²
+
² ²
= 0
subject to the boundary conditions u(0,y) = u(l,y) = u(x,0) = 0 , u(x,a) = Sin Ques‐6 A thin rectangular plate whose surface is imprevious to heat flow has at t=0 an arbitrary distribution of temperature f(x,y).Its four edges x=0,x=a,y=0,y=b are kept at zero temperature.Determine the temperature at a point of a plate as ‘t’ increases.Discuss the problem when
F(x,y) =
)
²
, find L(cos²at)
Ques‐3 If f(t) =
, find the Laplace transform of f(t).
Ques – 4 Given L 2
=
/
, Show that L (
√
) = √
Ques‐5 Find the Laplace Transform of t e‐t Sin2t Ques‐6 Using Laplace Transform , evaluate the following integrals:‐ √
(i) dt (ii) dt Ques‐7 Express the Following function in terms of Unit Step Function and find its Laplace Transform:‐ F(t) 1 1 2 t Ques‐8 Find the Laplace Transform of:‐ a) (t‐1)² u (t‐1) b) Sint u (t‐π) Ques‐9 Draw the graph and find the Laplace Transform of the triangular Wave function of Period 2c is given by:‐ F(t) = t ,0
F(t) = 1 , 0
‐1 , Ques‐11 Find the Laplace Transform of the periodic function :‐ F(t)= , for 0 ; F(t+T) = F(t) Ques‐12 (i) Find the inverse Laplace Transform of b) F(p) = a) F(p) = ²
Ques‐13 Use Convolution Theorem ,to evaluate ,
(i) L‐1 {
(ii) L‐1 {
Ques‐14 (i) Solve the Simultaneous Equation by Laplace Transform :‐ , + x = Sint ; Given x(0)=1 , y(0)=0 (ii) Solve the Simultaneous Equation by Laplace Transform :‐ ( D2 – 3) x – 4y =0 x + (D²+1)y =0 For t>0 , given that x=y= 0 = 2 at t = 0
²
UNIT‐V Ques‐1 Use the method of separation of variables , to solve the Equation:‐ 2
(i)
(ii)
² ²
+ u ,given that u(x,0)= 6 2
+ = 0
Ques‐2 A String is stretched and fastened to two points ‘l’ apart. Motion is started by displacing the string in the form, Y= A Sin , from which it is released at time t=0 .Show that the displacement of any point at a distance 'x’ from one end at time ‘t’ is given by, Y(x,t) = A Sin Cos Ques‐3 Show that how the Equation ²
²
c² = ² ² can be solved by the method of separation of variables. If the initial displacement and velocity of a string stretched between x=0 and x=l given by :‐ y=f(x) and = g(x) , determine the constants in the series solution. Ques‐4 Find the temperature in a bar of length 2 whose ends are kept at zero and lateral surface insulated if the initial temperature is given as:‐ Sin + 3 Sin Ques‐5 Use the separation of variables method to solve the equation:‐
² ²
+
² ²
= 0
subject to the boundary conditions u(0,y) = u(l,y) = u(x,0) = 0 , u(x,a) = Sin Ques‐6 A thin rectangular plate whose surface is imprevious to heat flow has at t=0 an arbitrary distribution of temperature f(x,y).Its four edges x=0,x=a,y=0,y=b are kept at zero temperature.Determine the temperature at a point of a plate as ‘t’ increases.Discuss the problem when
F(x,y) =
)
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