Differential Equations and Vector Calculus
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Differential Equations and Vector Calculus
English 18 leveled MCQs Free
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10 marks Long answer
Solve $(D^2 - 4D)y = e^x + \sin 3x \cos 2x$.
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Use the complementary function and particular integral to solve.
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10 marks Long answer
Solve the following equation by the method of variation of parameters $(D^2 + 3D + 2)y = e^x + x^2$.
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Apply the method of variation of parameters to find the solution.
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10 marks Long answer
Solve $(1 + 2x)^2 \frac{d^2y}{dx^2} - 6(1 + 2x) \frac{dy}{dx} + 16y = 8(1 + 2x)^2$.
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Use substitution to simplify and solve the differential equation.
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10 marks Long answer
In an L-C-R circuit, the charge $q$ on a plate of a condenser is given by $L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = E \sin pt$. The circuit is tuned to resonance so that $p^2 = \frac{1}{LC}$. If initially the current $i$ and the charge $q$ be zero, show that, for small values of $R/L$, the current in the circuit at time $t$ is given by $\frac{Et}{2L} \sin pt$.
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Use the method of undetermined coefficients to derive the expression.
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10 marks Long answer
Solve by the method of separation of variables $u_x = 2u_t + u$ where $u(x,0) = 6e^{-3x}$.
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Separate variables and solve the resulting ODEs.
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10 marks Long answer
Verify Gauss's divergence theorem for $\vec{F} = (x^2 - yz)\bar{i} + (y^2 - zx)\bar{j} + (z^2 - xy)\bar{k}$ take over the rectangular parallelepiped $0 \le x \le a, 0 \le y \le b, 0 \le z \le c$.
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Calculate the surface integral and volume integral to verify.
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10 marks Long answer
Verify Stokes' theorem for the vector field $\vec{F} = (2x - y)\bar{i} - yz^2\bar{j} - y^2z\bar{k}$ over the upper half surface of $x^2 + y^2 + z^2 = 1$ bounded by its projection on the xy-plane.
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Calculate the line integral and surface integral to verify.
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8 marks Long answer
Find the constant a, b and c such that the vector field defined by $\vec{F} = (4xy + az^3)\bar{i} + (bx^2 + 3z)\bar{j} + (6xz^2 + cy)\bar{k}$ is irrotational. With these values of a, b and c determine a scalar function $\phi$ such that $\vec{F} = \nabla\phi$.
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Set curl $\nabla \times \vec{F} = 0$ and solve for constants.
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4 marks Medium answer
Solve $p(1+q) = qz$.
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Use the method of characteristics to find the solution.
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4 marks Medium answer
Solve $x(z^2 - y^2)p + y(x^2 - z^2)q = z(y^2 - x^2)$.
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Apply the method of characteristics to solve the PDE.
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4 marks Medium answer
Find the directional derivative of $\phi(x,y,z) = xy + yz + zx$ in the direction of $-2\bar{i} + \bar{j} + 2\bar{k}$ at the point $(1, 2, 0)$.
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Calculate $\nabla \phi \cdot \hat{u}$ where $\hat{u}$ is the direction vector.
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4 marks Medium answer
Find the angle between the surfaces $x^2 + y^2 + z^2 = 12$ and $x^2 + y^2 - z = 12$ at $(2,2,2)$.
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Use the gradients to find the angle between the normals.
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2 marks Short answer
Evaluate $\frac{1}{D^2 - 4D + 4} xe^{2x}$.
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Apply the inverse operator to find the solution.
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2 marks Short answer
Solve the Euler's equation $x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y = 0$.
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Use the substitution $y = x^m$ to find the roots.
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2 marks Short answer
Find the general solution of $p + q = pq$.
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The general solution is $y = Ce^{x}$.
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2 marks Short answer
Prove that $\nabla . \bar{r} = 3$.
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Use the definition of divergence.
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2 marks Short answer
State Green's theorem.
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It relates the line integral around a simple closed curve to a double integral over the plane region.
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2 marks Short answer
Prove that $\nabla \left(\frac{\bar{r}}{r^3}\right) = 0$.
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Use the product and chain rules of differentiation.
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