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Differential Equations — Class 12 solved problems

29 problems from this chapter, each solved step by step.

  1. Solve dy/dx + y*tan(x) = sec(x) given y(0) = 1.

    y = sin(x) + cos(x)

  2. General solution of differential equation y log y dx - x dy = 0 is:

    x = A log y

  3. (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.

    e^y = e^x + (x³)/(3) + C

  4. 29. (a) Find the general solution of the differential equation 2x² (dy)/(dx) = y² + 2xy. (differential equation, general solution, calculus, homogeneous equatio

    The general solution of the differential equation is y = (2x)/(C - ln|x|).

  5. If y=3 e² x+2 e³ x, prove that (d² y)/(d x²)-5 (d y)/(d x)+6 y=0.

    The given equation (d² y)/(d x²)-5 (d y)/(d x)+6 y=0 is proven to be true.

  6. 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

    (5√3)/(2)

  7. 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:

    36

  8. 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:

    y(1) = e^(-4) - e^(-5)

  9. Let f be a differentiable function such that 2(x + 2)² f(x) - 3(x + 2)² = 10 ∫_0^x (t + 2) f(t) dt, x ≥ 0. Then f(2) is equal to

    19

  10. Solve the differential equation dy/dx + y*cot(x) = 2*x + x²*cot(x) given y(pi/2) = 0.

    y = x² - (π²)/(4)csc(x)

  11. Let a curve y = f(x) pass through the points (0, 5) and (log_e 2, k). If the curve satisfies the differential equation 2(3 + y) e²x dx - (7 + e²x) dy = 0, then

    k=8

  12. Let y=f(x) be the solution of the differential equation (dy)/(dx) + (xy)/(x²-1) = (x^4 + 4x)/(√(1-x²)), -1 < x < 1 such that f(0)=0. If 6 ∫_-1/2^(1/2) f(x) dx =

    27

  13. (b) Find a particular solution of the differential equation (x+1) (dy)/(dx) = 2e^(-y) - 1, given that y=0 when x=0. (differential equation, particular solution,

    y = ln|(2x + 1)/(x+1)|

  14. Let y = y(x) be the solution of the differential equation (x² + 1)y' - 2xy = (x^4 + 2x² + 1)cos x, with y(0) = 1. Then ∫_-3³ y(x) dx is:

    24

  15. Let y = y(x) be the solution of the differential equation 2 cos x (dy)/(dx) = sin 2x - 4y sin x, x (0, (π)/(2)). If y((π)/(3)) = 0, then y'((π)/(4)) + y((π)/(4)

    (5√2)/(4)

  16. Let y = y(x) be the solution of the differential equation (dy)/(dx) + 2y sec² x = 2sec² x + 3 tan x · sec² x, such that y(0) = (5)/(4). Then 12 ( y((π)/(4)) - e

    15 + 6 e^(-2)

  17. 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 start

    The speed with which the man can row the boat in still water is 9 km/h.

  18. Let x = x(y) be the solution of the differential equation y = (x - y (dx)/(dy)) sin ((x)/(y)), y > 0 and x(1) = (π)/(2). Then cos (x(2)) is equal to:

    2(ln 2)² - 1

  19. A radioactive substance decays according to the equation dN/dt = -kN. If the initial amount is N₀ and half-life is T, express N(t) in terms of N₀, T, and t.

    N(t) = N_0 2^(-t/T)

  20. Let y=y(x) be the solution of the differential equation (dy)/(dx) + 3(tan²x) y + 3y = sec²x, with y(0)=(1)/(3)+e³. Then y ((π)/(4)) is equal to:

    (4)/(3)

  21. Find the general solution of: y'' - 4y' + 4y = e^(2x)/x² using variation of parameters.

    y = c_1e²x + c_2xe²x - e²xln|x| - e²x

  22. Let y = y(x) be the solution of the differential equation cos x (log_e (cos x))² dy + (sin x - 3y sin x log_e (cos x)) dx = 0, x (0, (π)/(2)). If y((π)/(4)) = (

    12 (log_e ((√3)/(2)))²

  23. Let y = y(x) be the solution curve of the differential equation x(x² + e^x)dy + (e^x(x - 2)y - x³)dx = 0, x > 0, passing through the point (1, 0). Then y(2) is

    (4)/(4 + e²)

  24. If a curve y=y(x) passes through the point (1,(π)/(2)) and satisfies the differential equation (7x^4cot y - e^xcsc y)(dx)/(dy) = x^5, x 1, then at x=2, the valu

    (e(2e - 1))/(128)

  25. Solve the differential equation: (x² - y²)dx + 2xy dy = 0 given that y(1) = 1.

    x² + y² = 2x

  26. Solve the differential equation: (1+x²)y'' + 2xy' - 2y = 0 given that y₁ = x is a known solution.

    y = A x + B (x² - 1)

  27. Let f: (0, ∞) → R be a function which is differentiable at all points of its domain and satisfies the condition x² f'(x) = 2x f(x) + 3, with f(1) = 4. Then 2 f(

    39

  28. If y=A sin x+B cos x, then prove that (d² y)/(d x²)+y=0.

    The proof is complete, as (d² y)/(d x²)+y=0.

  29. If x=f(y) is the solution of the differential equation (1+y²) + (x - 2e^(tan^(-1) y)) (dy)/(dx) = 0, y (-(π)/(2), (π)/(2)) with f(0)=1, then f((1)/(√3)) is equa

    e^((π)/(6))

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