Bihar Board Class 12 (Commerce) Differential Equations — practice questions
78 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice Bihar Board Class 12 (Commerce) Differential Equations in the app →The ORDER and DEGREE of the DE (d²y/dx²)³ + (dy/dx)⁵ + y = 0 are:The general solution of a 3rd-order DE contains how many arbitrary constants?The DE dy/dx = y/x is best solved by:For the linear DE dy/dx + (1/x)y = x, the INTEGRATING FACTOR μ(x) is:If a population grows exponentially as dN/dt = 0.1 N (with t in years), the DOUBLING TIME is:By Newton's law of cooling, a hot tea cup at 90°C in a room at 25°C cools so that:A mass on a spring obeys d²x/dt² + ω²x = 0 (SHM). The period of oscillation is:Which kind of equation has a function, not a number, as its solution?Derivatives in several variables occur in which equation type?For y'' + sin(y') = 0, which statement about order and degree is correct?When is the degree of a differential equation defined?Find the order and degree of [1 + (y')^2]^(3/2) = y''.Which differential equation has y = e^(-3x) as a solution?Which function satisfies x (dy/dx) = 2y?What is a solution of a differential equation that contains arbitrary constants called?A solution obtained by giving particular values to the arbitrary constants is called what?How many arbitrary constants are in the general solution of a fourth order equation?The curve y = f(x) that satisfies a differential equation is called what?The variable separable method applies when terms in y can be gathered with which differential?The general solution of dy/dx = y/x (x, y positive) is which of these?Find the curve through (0, 2) whose slope at every point (x, y) is 3x^2.Find the curve through (1, 1) given x dy = (2x^2 + 1) dx, with x > 0.Find the curve through (0, 2) satisfying dy/dx = x/y.Find y when dy/dx = y and y = 3 at x = 0.At 5 percent continuous interest, Rs 1000 deposited grows to what after 10 years?With interest at r percent compounded continuously, the doubling time in years is what?A quantity whose rate of growth is proportional to its size follows which law?A function F(x, y) is homogeneous of degree n if F(lambda x, lambda y) equals what?What is the degree of homogeneity of (x^3 + y^3)/(xy)?Which equation is a homogeneous differential equation?Which substitution is used to solve dy/dx = F(y/x)?For a homogeneous equation written as dx/dy = F(x/y), which substitution is convenient?Solve dy/dx = y/x + tan(y/x) for the general solution.In dy/dx + Py = Q, a first order linear form, P and Q are what?For dy/dx + Py = Q, the integrating factor is which expression?For a linear equation dx/dy + P1 x = Q1, what must P1 and Q1 be?Find the integrating factor of y dx + (x - y^3) dy = 0, treating x as dependent.The date of birth of differential equations is taken as 11 November of which year?Who first used the word solution for a differential equation, in 1774?Who coined the old word integral for what is now called a solution, in 1690?What do derivatives in a differential equation usually stand for?What does a partial differential equation contain?Conditions given at different values of the independent variable are usually called what?Which is an example of a fourth order partial differential equation?Which equation gives a derivative in terms of earlier values?A difference equation differs from a differential equation because its independent variable takes what values?A differential equation is (d²y/dx²)³ + (dy/dx)⁴ + y = 0. What are its order and degree?For (d³y/dx³)² + (dy/dx)⁵ = x, find the order and degree.What is the order of the differential equation of the family y = A sin x + B cos x?Eliminating the constant a from y = a e^(2x) gives which differential equation?The family of straight lines y = Ax through the origin has which differential equation?What is the integrating factor of x dy/dx - y = x³ after writing it in standard form?Solving dy/dx = -x/y by separating variables givesThe general solution of dy/dx = 3x² is y = x³ + c. The particular solution with y = 2 at x = 0 isBacteria grow at a rate proportional to their number and double in 3 hours. When will they be 8 times?A population grows with dP/dt = kP and triples in 2 hours. How long will it take to become 9 times?In the growth solution P = c e^(kt), what does the constant c represent?A radioactive sample has a half life of 5 days. Of 800 mg, how much remains after 15 days?A radioactive element has a half life of 10 years. How much of a 640 g sample remains after 20 years?Why is there a negative sign in dx/dt = -kx for radioactive decay?For x = x0 e^(-kt), what fraction of x0 remains at time t = 1/k?A body at 60°C cools in a room at 20°C. By Newton's law its rate of cooling is proportional toWhich equation expresses Newton's law of cooling for body temperature θ and surroundings θ0?Equations with partial derivatives are called what?In an ordinary differential equation, the unknown function depends on how many real variables?An equation with only first-order derivatives of the unknown function is of which order?A vibrating string fixed at two endpoints leads to which type of problem?For a second-order ODE describing a moving particle, typical initial conditions are which pair?A linear differential equation is homogeneous when each term has what?Which numerical approach approximates a differential equation by a difference equation?In a difference equation, the independent variable takes what kind of values?Which theorem gives existence of a first-order solution?The harmonic oscillator equation approximates which nonlinear equation for small amplitudes?A derivative depending on earlier function values gives which equation?An equation whose unknown is a random process is called what?Which equation mixes derivatives with integrals of the unknown?Newton's laws give a differential equation for which unknown?The thin-film equation, an exception among natural laws, is of which order?