Differential Equations — Chapter Test | Class 12 IMO

Q1

In a culture, the bacteria count is 100,000. The number is increasing at a rate proportional to the number present. In 2 hours, it increases by 10%. The DE is dN/dt = kN. What is N at time t=0?

The initial count at t=0 is explicitly given in the problem as 100,000.

Q2

Solution of the differential equation (x + y) dy − dx = 0 is:

dx/dy = x + y → dx/dy − x = y. This is a linear DE in x. IF = e^(∫−1 dy) = e⁻ʸ. Solution: x(e⁻ʸ) = ∫y e⁻ʸ dy + C. Integrating by parts: −y e⁻ʸ − e⁻ʸ + c. Multiply by eʸ: x = −y − 1 + C eʸ, which is x + y + 1 = C eʸ.

Q3

The solution of dy/dx=(x+y)/x can be obtained using:

RHS depends only on y/x.

Q4

The solution of dy/dx = (y+1) is:

dy/(y+1)=dx. Hence ln(y+1)=x+C.

Q5

Find the general solution of the differential equation x dy + y dx = 0.

x dy = −y dx → dy/y = −dx/x. Integrating gives log y = −log x + log C → log(xy) = log C → xy = C.

Q6

Which method is most appropriate to solve dy/dx = (x + y)²?

The equation is not homogeneous or linear, and variables cannot be separated directly. Letting x + y = v gives 1 + dy/dx = dv/dx, so dv/dx = v² + 1, which reduces it to variable separable form.

Q7

The order of the differential equation representing the family of curves y = a sin(x + b), where a and b are arbitrary constants, is:

There are two independent arbitrary constants (a and b) in the given family of curves. Hence, the differential equation will be of order 2.

Q8

A radioactive substance follows dN/dt = −0.2N. The amount decreases because:

Negative proportionality constant implies decay.

Q9

A bank account grows continuously according to dA/dt = 0.08A. If A(0)=₹5000, then after time t the amount is:

Continuous growth follows A=A₀e^(kt).

Q10

The integrating factor (I.F.) of the linear differential equation dy/dx + Py = Q is given by:

By definition, for a first-order linear differential equation in y, dy/dx + Py = Q, the integrating factor is e^(∫P dx).

Q11

Rahul deposits ₹10,000 in a bank that pays interest compounded continuously at a rate of r% per annum. The differential equation for the principal P is:

Continuous compounding means the rate of change of principal is proportional to the principal. dP/dt = (r/100)P.

Q12

The differential equation dy/dx = y is:

It can be written as dy/y = dx, hence variable separable.

Q13

The solution of dy/dx = eˣ is:

Integrating eˣ gives eˣ.

Q14

The order of the differential equation d⁴y/dx⁴ + (d²y/dx²)² = 0 is:

Highest derivative present is fourth order.

Q15

The appropriate substitution to solve the homogeneous differential equation dx/dy = f(x/y) is:

When the equation is expressed as dx/dy = f(x/y), it is easier to substitute x = vy, which gives dx/dy = v + y(dv/dy).