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HPCET Differential Equations — practice questions — page 2

Explore 73 free Differential Equations questions in HPCET Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Differential Equations questions and exercises with answers

Showing questions 31–60 of 73.

Order and degree of (d²y/dx²)² + y = 0:Read the answer and worked solution →Homogeneous DE dy/dx = f(y/x): solved by substitutionRead the answer and worked solution →Solve dy/dx = 1 + y²:Read the answer and worked solution →For exponential decay (radioactive): dN/dt = -λN. Solution:Read the answer and worked solution →Solve dy/dx = (x + y)/(x - y) (homogeneous, deg 0):Read the answer and worked solution →Solve dy/dx + y = sin x:Read the answer and worked solution →Solve d²y/dx² + y = 0 (SHM):Read the answer and worked solution →Solve d²y/dx² - 4 dy/dx + 4y = 0:Read the answer and worked solution →For population growth with carrying capacity (logistic): dP/dt = rP(1 - P/K), the limiting value as t → ∞:Read the answer and worked solution →For RC circuit charging through resistor R: dq/dt + q/(RC) = V/R. Time constant:Read the answer and worked solution →Solve x dy/dx + y = x²:Read the answer and worked solution →Initial value problem: dy/dx = e^x with y(0) = 2. Solution:Read the answer and worked solution →Order of DE obtained by eliminating constants A, B from y = A e^x + B e^(-x):Read the answer and worked solution →Solve dy/dx = y² (Bernoulli-like):Read the answer and worked solution →For Newton's law of cooling, body at 80°C cools in room at 20°C. After time, T = 60°C. Find k if time = 5 min:Read the answer and worked solution →For DE M(x,y) dx + N(x,y) dy = 0 to be EXACT:Read the answer and worked solution →Linear independence: two solutions y₁, y₂ of 2nd-order linear DE are independent iff Wronskian W:Read the answer and worked solution →For DE y'' + 4y = 0, general solution:Read the answer and worked solution →Complete the definition of order and degreeRead the answer and worked solution →True or false: order and degreeRead the answer and worked solution →General solution or particular solution?Read the answer and worked solution →Match each method to the form it solvesRead the answer and worked solution →Recognising the typeRead the answer and worked solution →Complete the definition of a differential equationRead the answer and worked solution →Order the steps: solve by separating variablesRead the answer and worked solution →Tap every statement true of a general solutionRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →

Worked practice questions

Order and degree of (d²y/dx²)² + y = 0:

  1. 1 and 2
  2. 2 and 2 (2nd-order, squared)
  3. Not defined
  4. 2 and 1
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Correct answer: B. 2 and 2 (2nd-order, squared)

Highest order derivative is d²y/dx² (order = 2). It's raised to power 2 (degree = 2). Both well-defined here since no radicals/fractions of derivatives.

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Homogeneous DE dy/dx = f(y/x): solved by substitution

  1. v = x²
  2. v = x
  3. v = y/x (so y = vx, dy/dx = v + x dv/dx)
  4. v = xy
Show answer and solution

Correct answer: C. v = y/x (so y = vx, dy/dx = v + x dv/dx)

For homogeneous DE (f depends only on y/x), substitute v = y/x. Reduces to variable separable form. Standard technique for homogeneous-of-degree-zero rate functions.

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Solve dy/dx = 1 + y²:

  1. y = tan(x + C) (using arctan integral)
  2. y = 1/(x + C)
  3. y = x + C
  4. y = e^x
Show answer and solution

Correct answer: A. y = tan(x + C) (using arctan integral)

**Separable:** dy/(1 + y²) = dx → arctan y = x + C → y = tan(x + C). Solutions blow up at x = π/2 - C (finite-time singularity).

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