COMPACT QUESTION (DIFFERENTIATION -ORDER-2)

EXERCISE -A

1) If y= cos²x - sin²x, find d²y/dx² at x=0.        -4

2) If f(x)= 3 sinx - 4sin³x, find f"(π/2).        9

3) If y²= mx², show d²y/dx²=0.


4) If f(x)= x²eˣ, then find f"(0).       2

5) If x= a cos2t and y= b sin²t, find d²y/dx².       0

6) For the parabola y²= 4ax, show d²y/dx² . d²x/dy²= 2a/y³.

7) If pvʸ = constant, show that, v² d²p/dv² = y(y+1)p.


EXERCISE -B

1) xᵃ yᵇ =(x+ y)ᵃ⁺ᵇ, show dy/dx is independent of a and b and hence show that d²y/dx²=0.

2) If y= (1+ x)ⁿ/(1- x)ⁿ, show d²y/dx²= 2(n+ x)/(1- x²). dy/dx.

3) If y= x tan⁻¹(x/y), then evaluate dy/dx and d²y/dx².

4) If log y= sin⁻¹x, prove that, (1- x²) d²y/dx² - x dy/dx =y

5) If y= sin(m sin⁻¹x)[or cos(m cos⁻¹x)], then she(1- x²) d²y/dx² - x dy/dx + m²y=0.

6) If y= a cos(logx)+ b sin(logx), show x² d²y/dx² + x dy/dx + y=0.

7) If x² + xy+ y² =1, show (x +2y)³ d²y/dx² +6=0.

8) If y² = ax² + 2bx+ c. Show d²x/dy² = (b² - ac)/(ax + b)³.

9) If y= A(x + √(x²-1))ⁿ + B(x - √(x²-1))ⁿ, show (1- x²) d²y/dx² - x dy/dx + n²y=0.

10) If x= cost and y= logt, then show at t=π/2 is d²y/dx² + (dy/dx)² =0.

11) If 2x = y¹⁾ᵐ + y⁻¹⁾ᵐ, show (1- x²) d²y/dx² - x dy/dx+ m²y=0.

12) If y³ - 3ax² + x³=0, then show d²y/dx² + 2a²x²/y⁵=0

13) If x= Sint and y= sin nt, show that (1- x²) d²y/dx² - x dy/dx + n²y=0.

14) If eˣ = yʸ, show (x+ y) d²y/dx² +(dy/dx)² =0.

15) If x= eᵗ sint and y= eᵗ cost, show (x+ y)² d²y/dx² = 2(x dy/dx - y).

16) If y= t² and x = cost (or sint), show that, (1- x²) d²y/dx² - x dy/dx =2.

17) If (a+ bx)eʸ⁾ˣ = x, show that x³ d²y/dx² = (x dy/dx - y)².

18) If y= Aeᵗ + Be⁻ᵗ and x= sint, show (1- x²) d²y/dx² - x dy/dx = y.

19) If x= eᵗ and d²y/dx² + p²y=0, show x² d²y/dx² + x dy/dx+ p²y=0.

20) If p² = a²cos²x + b² sin²x, show p+ d²p/dx²= a²b²/p³.

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