Exercise 9.2 answers: Differential Equations

Open NCERT p. 306One way to think about it

1. Differentiate y = eˣ + 1 once: y′ = eˣ.
2. Differentiate again: y′′ = eˣ.
3. Put these into y′′ − y′: eˣ − eˣ = 0, which matches the right side.

In shortYes — substituting the function and its derivative(s) into the left side gives exactly the right side, so it is a solution.

Watch this explained “Testing a candidate”, 7:39 into What makes an equation differential, and why the answer is a curve not a number

Open NCERT p. 306One way to think about it

1. Differentiate y = x² + 2x + C: y′ = 2x + 2 (the constant C disappears).
2. Put this into y′ − 2x − 2: (2x + 2) − 2x − 2 = 0, which matches the right side.

In shortYes — substituting the function and its derivative(s) into the left side gives exactly the right side, so it is a solution.

Watch this explained “Testing a candidate”, 7:39 into What makes an equation differential, and why the answer is a curve not a number

Open NCERT p. 306One way to think about it

1. Differentiate y = cos x + C: y′ = −sin x.
2. Put this into y′ + sin x: −sin x + sin x = 0, which matches the right side.

In shortYes — substituting the function and its derivative(s) into the left side gives exactly the right side, so it is a solution.

Watch this explained “Testing a candidate”, 7:39 into What makes an equation differential, and why the answer is a curve not a number

Open NCERT p. 306One way to think about it

1. Differentiate y = √(1 + x²) with respect to x
2. y′ = (1/2)(1 + x²)^(−1/2) · 2x = x / √(1 + x²)
3. Compute the right side: xy / (1 + x²) = x · √(1 + x²) / (1 + x²)
4. = x / √(1 + x²)
5. Left side: y′ = x / √(1 + x²)
6. Right side: xy / (1 + x²) = x / √(1 + x²)
7. Both sides are equal, so the function satisfies the differential equation

In shortThe function y = √(1 + x²) satisfies the differential equation y′ = xy / (1 + x²)

Watch this explained “Testing a candidate”, 7:39 into What makes an equation differential, and why the answer is a curve not a number

Open NCERT p. 306One way to think about it

1. Differentiate y = Ax: y′ = A.
2. Put this into xy′: x × A = Ax, which is exactly y.

In shortYes — substituting the function and its derivative(s) into the left side gives exactly the right side, so it is a solution.

Watch this explained “Testing a candidate”, 7:39 into What makes an equation differential, and why the answer is a curve not a number

Open NCERT p. 306One way to think about it

1. y = x sin x, so by the product rule y′ = sin x + x cos x.
2. Left side: xy′ = x sin x + x² cos x.
3. Right side: x² − y² = x² − x² sin²x = x²(1 − sin²x) = x² cos²x, so √(x² − y²) = √(x² cos²x) = |x cos x|.
4. The given condition (x > y or x < −y) makes x² − y² > 0, so this root is real and not zero. On an interval where x cos x > 0 (for example 0 < x < π/2), |x cos x| = x cos x.
5. There the right side is y + x√(x² − y²) = x sin x + x · x cos x = x sin x + x² cos x, which equals the left side.

In shortWhere x cos x > 0 (for example 0 < x < π/2), both sides equal x sin x + x² cos x, so y = x sin x is a solution there. Note: √(x² cos²x) is |x cos x|, so where x cos x < 0 the two sides differ — at x = π the left side is −π² and the right side is π².

Watch this explained “Testing a candidate”, 7:39 into What makes an equation differential, and why the answer is a curve not a number

Open NCERT p. 306One way to think about it

1. Differentiate xy = log y + C with respect to x, treating y as a function of x: y + xy′ = y′/y.
2. Multiply every term by y: y² + xyy′ = y′.
3. Collect the y′ terms: y² = y′ − xyy′ = y′(1 − xy).
4. Divide by (1 − xy), which is allowed because xy ≠ 1: y′ = y²/(1 − xy).

In shortDifferentiating xy = log y + C gives y′ = y²/(1 − xy), which is exactly the given differential equation, so xy = log y + C is a solution.

Watch this explained “Testing a candidate”, 7:39 into What makes an equation differential, and why the answer is a curve not a number

Open NCERT p. 306One way to think about it

1. Differentiate y − cos y = x with respect to x: y′ + y′ sin y = 1, so y′ = 1/(1 + sin y).
2. In the left side of the equation, replace x by y − cos y: y sin y + cos y + (y − cos y) = y sin y + y = y(sin y + 1).
3. Multiply by y′: y(sin y + 1) × 1/(1 + sin y) = y, which matches the right side.

In shortYes — substituting the function and its derivative(s) into the left side gives exactly the right side, so it is a solution.

Watch this explained “Testing a candidate”, 7:39 into What makes an equation differential, and why the answer is a curve not a number

Open NCERT p. 306One way to think about it

1. The relation is x + y = tan⁻¹y.
2. Differentiate both sides with respect to x: 1 + y′ = y′/(1 + y²).
3. Multiply both sides by (1 + y²): (1 + y²) + (1 + y²)y′ = y′.
4. Expand the left side: 1 + y² + y′ + y²y′ = y′.
5. Take y′ away from both sides: 1 + y² + y²y′ = 0.
6. Rearranged, this is y²y′ + y² + 1 = 0, exactly the given equation.
7. So the relation satisfies the equation, and it is a solution.

In shortYes. Differentiating x + y = tan⁻¹y gives y²y′ + y² + 1 = 0, so it is a solution.

Watch this explained “Testing a candidate”, 7:39 into What makes an equation differential, and why the answer is a curve not a number

Open NCERT p. 306One way to think about it

1. The function is y = √(a² − x²).
2. Differentiate: dy/dx = −x / √(a² − x²) = −x/y.
3. Multiply both sides by y: y × dy/dx = −x.
4. Rearrange: x + y × dy/dx = 0 — exactly the given equation (for y ≠ 0).
5. Both sides agree, so the function is a solution.

In shortYes — y = √(a² − x²) satisfies x + y(dy/dx) = 0 for y ≠ 0.

Watch this explained “Testing a candidate”, 7:39 into What makes an equation differential, and why the answer is a curve not a number

Open NCERT p. 306Matches NCERT’s answer

1. A general solution keeps arbitrary constants — it has not been pinned down to one curve yet.
2. The rule: the number of arbitrary constants in a general solution equals the order of the equation.
3. Here the order is 4 (given), so the general solution carries 4 arbitrary constants.
4. That matches option (D).

Answer(D) 4

Watch this explained “The climb, on four letters”, 6:23 into General against particular: why the arbitrary constants are counted by the order

Open NCERT p. 306Matches NCERT’s answer

1. A particular solution is a general solution with every constant already given a fixed value.
2. So a particular solution never carries any arbitrary constant left over — the count is 0.
3. This is true whatever the order of the equation, so the order (third) does not change the answer.
4. That matches option (D).

Answer(D) 0

Watch this explained “The case that comes apart”, 10:15 into General against particular: why the arbitrary constants are counted by the order