JNTUH B.Tech - R25, R22 - Ordinary Differential Equations and Vector Calculus - Important Questions

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JNTUH B.Tech - R25, R22 - Ordinary Differential Equations and Vector Calculus - Important Questions

Unit 1 : First Order Ordinary Differential Equations

• Solve exact differential equations and verify whether a given differential equation is exact or not.
• Solve differential equations reducible to exact differential equations using integrating factors.
• Solve first-order linear differential equations.
• Solve Bernoulli’s differential equations.
• Find orthogonal trajectories of a family of curves in Cartesian coordinates.
• State and solve applications based on Newton’s law of cooling.
• Solve problems based on the law of natural growth.
• Solve problems based on the law of natural decay.
• Distinguish between exact and non-exact differential equations and explain the conditions for exactness.
• Solve real-world application problems involving first-order ordinary differential equations.

Unit 2 : Ordinary Differential Equations of Higher Order

• Solve second order linear differential equations with constant coefficients.
• Solve non-homogeneous differential equations when the right-hand side is of the form eax.
• Solve non-homogeneous differential equations when the right-hand side is of the form sinax or cosax.
• Solve non-homogeneous differential equations when the right-hand side is a polynomial in x.
• Solve differential equations with non-homogeneous terms of the form eaxV(x) and xV(x).
• Solve second order differential equations using the method of variation of parameters.
• Solve Cauchy-Euler differential equations.
• Solve Legendre’s differential equations reducible to linear ODEs with constant coefficients.
• Compare the method of undetermined coefficients and variation of parameters.
• Solve application problems involving electric circuits using differential equations.

Unit 3 : Laplace Transforms

• Find the Laplace transforms of standard functions.
• Apply the first shifting theorem to find Laplace transforms and inverse Laplace transforms.
• Apply the second shifting theorem using the unit step function.
• Find the Laplace transform involving the Dirac delta function.
• Find Laplace transforms of functions multiplied or divided by t.
• Find Laplace transforms of derivatives and integrals of functions.
• Evaluate definite integrals using Laplace transforms.
• Find the Laplace transform of periodic functions.
• Find inverse Laplace transforms by partial fractions, convolution, and other standard methods.
• Solve initial value problems using the Laplace transform method.

Unit 4 : Vector Differentiation

• Define scalar point functions and vector point functions with suitable examples.
• Find the gradient of a scalar function and explain its geometrical significance.
• Find the divergence of a vector field and interpret its physical meaning.
• Find the curl of a vector field and explain its significance.
• Evaluate directional derivatives and determine the maximum rate of change.
• Find the equation of the tangent plane and normal line to a given surface.
• Verify important vector identities involving gradient, divergence, and curl.
• Determine the scalar potential function for a given conservative vector field.
• Test whether a vector field is solenoidal or irrotational.
• Solve problems involving scalar potential functions, solenoidal vectors, and irrotational vectors.

Unit 5 : Vector Integration

• Evaluate line integrals of scalar and vector functions.
• Evaluate surface integrals over given surfaces.
• Evaluate volume integrals over specified regions.
• State Green’s theorem and apply it to evaluate line integrals.
• State Gauss divergence theorem and apply it to evaluate surface integrals.
• State Stokes’ theorem and apply it to evaluate line and surface integrals.
• Convert a line integral into a double integral using Green’s theorem.
• Convert a surface integral into a volume integral using Gauss divergence theorem.
• Convert a line integral into a surface integral using Stokes’ theorem.
• Solve applications involving Green’s, Gauss, and Stokes’ theorems.