HPAS Maths Optional PYQs: Calculus of Variations

On that page, you will find year-wise and question-wise solutions.

Based on the sources provided, here is a compilation of all questions pertaining to the Calculus of Variations topic.

HPAS 2024 Calculus of Variations Questions

Question 6(a): Is the Jacobi condition fulfilled for the extremal of the functional: $$ \mathbf{\int_{0}^{a} (y’^2 – 4y^2 – e^{-x^2}) dx} $$ with fixed boundaries $A(0,0)$ and $B(a,0)$? (where $y’ = dy/dx$). (Note: $a \ne \frac{n\pi}{2}$ is given in the context of the question).

HPAS 2023 Calculus of Variations Questions

Question 7(a): On which curve can the functional: $$ \mathbf{\int_{0}^{\pi/2} (y’^2 – y^2 + 2xy) dx} $$ with boundary conditions $\mathbf{y(0)=0}$ and $\mathbf{y(\frac{\pi}{2})=0}$, be extremized? (where $y’ = \frac{dy}{dx}$).

HPAS 2021 Calculus of Variations Questions

Question 6(c): Test for an extremum of the functional: $$ \mathbf{I[y(x)] = \int_{0}^{1} (xy + y^2 – 2y^2 y’) dx} $$ with boundary conditions $\mathbf{y(0) = 1}$ and $\mathbf{y(1) = 2}$, where $y’ = \frac{dy}{dx}$.

HPAS 2020 Calculus of Variations Questions

Question 6(a): Find the shortest distance between the parabola $\mathbf{y=x^2}$ and the straight line $\mathbf{x-y=5}$.

Question 8(b): Find the extremal of the following functional: $$ \mathbf{v[y(x)] = \int_{0}^{1} (2y + (y”)^2) dx} $$ that satisfies the conditions $\mathbf{y(0)=0}$, $\mathbf{y'(0)=1}$, $\mathbf{y(1)=1}$, and $\mathbf{y'(1)=1}$.

HPAS 2019 Calculus of Variations Questions

Question 8(b): Using the Euler equation, find the extremal of the following functional: $$ \mathbf{\int_{a}^{b} (12xy(x) + (y’)^2) dx} $$

HPAS 2017 Calculus of Variations Questions

Question 1(d): On which curves can the functional: $$ \mathbf{H[y(x)] = \int_{0}^{1} [(y’)^2 + 12xy] dx} $$ with boundary conditions $\mathbf{y(0)=0}$ and $\mathbf{y(1)=1}$, be extremized?.

Question 8(a): Find the path on which a particle, in the absence of friction, will slide from one fixed point to another point not in the same vertical line in the shortest time under the action of gravity (The Brachistochrone problem).

HPAS 2016 Calculus of Variations Questions

Question 2(a): On which curve can the functional: $$ \mathbf{\int_{0}^{\pi/2}(y^{\prime2}-y^{2}+2xy)dy} $$ with $\mathbf{y(0)=0}$ and $\mathbf{y(\pi/2)=0}$ be extremized?. (Note: The variable of integration appears as $dy$ in the source, but it is implied to be $dx$ for a functional in $y(x)$).

HPAS 2015 Calculus of Variations Questions

Question 7(b): Find the extremum curve for the functional: $$ \mathbf{I[y(x)]=\int_{0}^{x_{2}}\sqrt{\frac{1+(y^{\prime})^{2}}{y}}dx} $$ given that: $\mathbf{y(0)=0}$ and $\mathbf{y_{2}=x_{2}+5}$.

HPAS 2014 Calculus of Variations Questions

Question 7(b): Find the extremal curve of the functional: $$ \mathbf{I[y(x),z(x)]=\int_{0}^{\pi/2}{(y^{\prime})^{2}+(z^{\prime})^{2}+2yz}dx} $$ given that: $\mathbf{y(0)=0}$, $\mathbf{y(\frac{\pi}{2})=-1}$; $\mathbf{z(0)=0}$, $\mathbf{z(\frac{\pi}{2})=1}$.