Differential Equations Basics Self Test

Differential Equations Quiz

Test your knowledge of ODEs, PDEs, and solution methods

Q1

A first-order ordinary differential equation is of the form:

$F(x, y, y', y'') = 0$ $F(x, y, y') = 0$ $F(x, y) = 0$ $F(y', y'') = 0$

Q2

The general solution to the separable differential equation $\frac{dy}{dx} = f(x)g(y)$ is:

$\int\frac{1}{g(y)}dy = \int f(x)dx + C$ $\int g(y)dy = \int f(x)dx + C$ $\int f(x)dx = \int g(y)dy + C$ $\int\frac{1}{f(x)}dx = \int\frac{1}{g(y)}dy + C$

Q3

An integrating factor for a first-order linear differential equation $\frac{dy}{dx} + P(x)y = Q(x)$ is:

$e^{\int P(x)dx}$ $e^{\int Q(x)dx}$ $e^{\int (P(x)+Q(x))dx}$ $e^{\int (P(x)Q(x))dx}$

Q4

The integrating factor for the differential equation $x\frac{dy}{dx} + 2y = x^2$ is:

$x$ $x^2$ $\frac{1}{x}$ $\frac{1}{x^2}$

Q5

The Wronskian of two functions $y_1$ and $y_2$ is defined as:

$W(y_1, y_2) = y_1y_2' - y_2y_1'$ $W(y_1, y_2) = y_1'y_2 - y_2'y_1$ $W(y_1, y_2) = y_1y_2' + y_2y_1'$ $W(y_1, y_2) = y_1'y_2' - y_1y_2$

Q6

If the Wronskian of two solutions of a second-order linear homogeneous differential equation is nonzero at some point in an interval, then the solutions are:

Linearly dependent on that interval Linearly independent on that interval Identical on that interval Not solutions on that interval

Q7

An initial value problem consists of:

A differential equation and an initial condition Only a differential equation Only an initial condition A differential equation and boundary conditions

Q8

For the initial value problem $\frac{dy}{dx} = 3x^2$, $y(0) = 1$, the solution is:

$y = x^3 + C$ $y = x^3$ $y = x^3 + 1$ $y = 3x^2 + 1$

Q9

A singular solution of a differential equation is:

A solution that cannot be obtained from the general solution by specifying values of arbitrary constants A solution that is identically zero A solution that contains singular points A solution that is not differentiable

Q10

For the differential equation $y = px + f(p)$, where $p = \frac{dy}{dx}$, the singular solution is obtained by:

Eliminating $p$ between $y = px + f(p)$ and $0 = x + f'(p)$ Setting $p = 0$ in the general solution Integrating with respect to $p$ Differentiating with respect to $x$

Q11

The P-discriminant method is used to find:

Singular solutions of first-order differential equations General solutions of first-order differential equations Particular solutions of first-order differential equations Solutions of partial differential equations

Q12

For the Clairaut's equation $y = px + f(p)$, the P-discriminant is obtained by:

Differentiating with respect to $p$ and setting the result to zero Differentiating with respect to $x$ and setting the result to zero Differentiating with respect to $y$ and setting the result to zero Integrating with respect to $p$

Q13

The C-discriminant method is used to find:

Envelope of a family of curves General solution of a differential equation Singular points of a differential equation Critical points of a differential equation

Q14

For a one-parameter family of curves $F(x, y, C) = 0$, the C-discriminant is obtained by:

Eliminating $C$ between $F(x, y, C) = 0$ and $\frac{\partial F}{\partial C} = 0$ Setting $C = 0$ in $F(x, y, C) = 0$ Differentiating $F$ with respect to $x$ and setting the result to zero Integrating $F$ with respect to $C$

Q15

The characteristic equation of the differential equation $y'' + ay' + by = 0$ is:

$r^2 + ar + b = 0$ $r^2 - ar + b = 0$ $r^2 + ar - b = 0$ $r^2 - ar - b = 0$

Q16

If the characteristic equation of a second-order linear homogeneous differential equation with constant coefficients has complex roots $\alpha \pm \beta i$, then the general solution is:

$y = C_1e^{\alpha x} + C_2e^{\beta x}$ $y = e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x))$ $y = C_1\cos(\alpha x) + C_2\sin(\beta x)$ $y = e^{\beta x}(C_1\cos(\alpha x) + C_2\sin(\alpha x))$

Q17

The total differential equation $Pdx + Qdy + Rdz = 0$ is integrable if:

$\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$, $\frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y}$, and $\frac{\partial R}{\partial x} = \frac{\partial P}{\partial z}$ $P(\frac{\partial Q}{\partial z} - \frac{\partial R}{\partial y}) + Q(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}) + R(\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x}) = 0$ $\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} = 0$ $\frac{\partial P}{\partial x} = \frac{\partial Q}{\partial y} = \frac{\partial R}{\partial z}$

Q18

For the total differential equation $yzdx + zxdy + xydz = 0$, the integrating factor is:

$\frac{1}{xyz}$ $xyz$ $\frac{1}{x+y+z}$ $x+y+z$

Q19

The necessary and sufficient condition for the total differential equation $Pdx + Qdy = 0$ to be exact is:

$\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$ $\frac{\partial P}{\partial x} = \frac{\partial Q}{\partial y}$ $\frac{\partial P}{\partial y} = -\frac{\partial Q}{\partial x}$ $\frac{\partial P}{\partial x} = -\frac{\partial Q}{\partial y}$

Q20

For the total differential equation $Pdx + Qdy + Rdz = 0$, the necessary and sufficient condition for integrability is:

$P(\frac{\partial Q}{\partial z} - \frac{\partial R}{\partial y}) + Q(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}) + R(\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x}) = 0$ $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$, $\frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y}$, and $\frac{\partial R}{\partial x} = \frac{\partial P}{\partial z}$ $\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} = 0$ $\frac{\partial P}{\partial x} = \frac{\partial Q}{\partial y} = \frac{\partial R}{\partial z}$

Q21

A first-order partial differential equation is of the form:

$F(x, y, z, \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}) = 0$ $F(x, y, z, \frac{\partial^2 z}{\partial x^2}, \frac{\partial^2 z}{\partial y^2}) = 0$ $F(x, y, \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}) = 0$ $F(x, y, z) = 0$

Q22

The general solution of a first-order partial differential equation $F(x, y, z, p, q) = 0$, where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$, contains:

One arbitrary function Two arbitrary functions One arbitrary constant Two arbitrary constants

Q23

Lagrange's method is used to solve:

First-order linear partial differential equations Second-order linear partial differential equations First-order nonlinear partial differential equations Second-order nonlinear partial differential equations

Q24

For the first-order linear partial differential equation $Pp + Qq = R$, where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$, Lagrange's auxiliary equations are:

$\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}$ $\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{0}$ $\frac{dx}{0} = \frac{dy}{Q} = \frac{dz}{R}$ $\frac{dx}{P} = \frac{dy}{0} = \frac{dz}{R}$

Q25

Charpit's method is used to solve:

First-order linear partial differential equations First-order nonlinear partial differential equations Second-order linear partial differential equations Second-order nonlinear partial differential equations

Q26

In Charpit's method for solving $F(x, y, z, p, q) = 0$, the auxiliary equations are:

$\frac{dx}{\frac{\partial F}{\partial p}} = \frac{dy}{\frac{\partial F}{\partial q}} = \frac{dz}{p\frac{\partial F}{\partial p} + q\frac{\partial F}{\partial q}} = \frac{dp}{-\frac{\partial F}{\partial x} - p\frac{\partial F}{\partial z}} = \frac{dq}{-\frac{\partial F}{\partial y} - q\frac{\partial F}{\partial z}}$ $\frac{dx}{\frac{\partial F}{\partial x}} = \frac{dy}{\frac{\partial F}{\partial y}} = \frac{dz}{\frac{\partial F}{\partial z}} = \frac{dp}{\frac{\partial F}{\partial p}} = \frac{dq}{\frac{\partial F}{\partial q}}$ $\frac{dx}{p} = \frac{dy}{q} = \frac{dz}{1} = \frac{dp}{0} = \frac{dq}{0}$ $\frac{dx}{\partial F/\partial x} = \frac{dy}{\partial F/\partial y} = \frac{dz}{\partial F/\partial z} = \frac{dp}{\partial F/\partial p} = \frac{dq}{\partial F/\partial q}$

Q27

A second-order partial differential equation of the form $A\frac{\partial^2 z}{\partial x^2} + B\frac{\partial^2 z}{\partial x\partial y} + C\frac{\partial^2 z}{\partial y^2} + D\frac{\partial z}{\partial x} + E\frac{\partial z}{\partial y} + Fz + G = 0$ is classified as:

Elliptic if $B^2 - 4AC < 0$ Parabolic if $B^2 - 4AC = 0$ Hyperbolic if $B^2 - 4AC > 0$ All of the above

Q28

The partial differential equation $\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = 0$ is:

Elliptic Parabolic Hyperbolic None of the above

Q29

For the differential equation $y'' + y = 0$ with initial conditions $y(0) = 0$ and $y'(0) = 1$, the Wronskian of the two linearly independent solutions $\sin(x)$ and $\cos(x)$ is:

$1$ $-1$ $0$ $\sin^2(x) + \cos^2(x)$

Q30

The partial differential equation $\frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial y^2} = 0$ can be transformed into:

$\frac{\partial^2 z}{\partial u\partial v} = 0$ by the change of variables $u = x + y$, $v = x - y$ $\frac{\partial^2 z}{\partial u^2} + \frac{\partial^2 z}{\partial v^2} = 0$ by the change of variables $u = x + y$, $v = x - y$ $\frac{\partial^2 z}{\partial u^2} - \frac{\partial^2 z}{\partial v^2} = 0$ by the change of variables $u = x + y$, $v = x - y$ $\frac{\partial^2 z}{\partial u\partial v} = 0$ by the change of variables $u = x$, $v = y$

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