SECTION B This section comprises 5 Very Short Answer (VSA) type questions of 2 marks each. 21. A man in a boat goes 12 km downstream and comes back to the starting point by rowing non-stop in a total time of 3 hours. If the speed of the stream is 3 km/h, find the speed with which the man can row the boat in still water. 22. (a) Solve the following differential equation: $$\frac{dy}{dx} = \frac{2-y}{x+1}$$ OR (b) If $$\int_a^b x^3 dx = 0$$ and $$\int_a^b x^2 dx = \frac{2}{3}$$ , then find the values of 'a' and 'b'. (differential equation, definite integral, speed, velocity, time)

Step 1: Define variables and speeds

First, we define the variables. Let $v_b$ be the speed of the boat in still water, which is what we need to find. The speed of the stream, $v_s$, is given as 3 km/h. When the boat travels downstream, its speed is the sum of the boat's speed and the stream's speed. When it travels upstream, its speed is the difference between the boat's speed and the stream's speed.

Step 2: Formulate time equations

The distance traveled downstream is 12 km, and the distance traveled upstream is also 12 km. Using the formula Time = Distance / Speed, we can write expressions for the time taken for the downstream journey ($t_{\text{down}}$) and the upstream journey ($t_{\text{up}}$).

Step 3: Set up total time equation

The problem states that the total time for the round trip (downstream and back upstream) is 3 hours. So, we sum the individual time expressions and set them equal to 3.

Step 4: Solve the quadratic equation

To solve for $v_b$, we first find a common denominator and simplify the equation. This leads to a quadratic equation. We then solve the quadratic equation by factoring or using the quadratic formula to find the possible values for $v_b$.

Step 5: Select the valid speed

Since speed cannot be negative, we discard the negative solution. Therefore, the speed of the man in still water is 9 km/h.

More from this chapter

• Solve dy/dx + y*tan(x) = sec(x) given y(0) = 1.
• General solution of differential equation y log y dx - x dy = 0 is:
• (a) Solve the following differential equation: (dy)/(dx) = e^(x - y) + x² e^(-y). OR (b) Solve the following differential equation: (x² - y²) dx + 2xy dy = 0.
• 29. (a) Find the general solution of the differential equation 2x² (dy)/(dx) = y² + 2xy. (differential equation, general solution, calculus, homogeneous equatio
• If y=3 e² x+2 e³ x, prove that (d² y)/(d x²)-5 (d y)/(d x)+6 y=0.
• Let y = y(x) be the solution of the differential equation (xy - 5x² √(1 + x²)) dx + (1 + x²) dy = 0, y(0) = 0. Then y(√3) is equal to
• Let f: R→ R be a thrice‐differentiable odd function satisfying f''(x)=f(x), f(0)=0, and f'(0)=3. Then 9f(ln 3) is equal to:
• Let f(x) = x - 1 and g(x) = e^x for x R. If (dy)/(dx) = ( e^(-2√x) g(f(f(f(x)))) - (y)/(√x)), y(0) = 0, then y(1) is:

All Differential Equations solutions →