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VITEEE Differential Equations — practice questions

Explore 51 free Differential Equations questions in VITEEE 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 1–30 of 51.

Order of differential equation:Read the answer and worked solution →General solution of dy/dx = k y:Read the answer and worked solution →Variable separable form:Read the answer and worked solution →For dy/dx = x, general solution:Read the answer and worked solution →Solve dy/dx = (y/x):Read the answer and worked solution →Linear DE of first order: dy/dx + P(x) y = Q(x). Solution uses:Read the answer and worked solution →Integrating factor for dy/dx + y = e^x:Read the answer and worked solution →Solve dy/dx = -y/x with y(1) = 2:Read the answer and worked solution →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 →

Worked practice questions

Order of differential equation:

  1. Degree
  2. Total of x
  3. Highest derivative present
  4. Coefficient of y
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Correct answer: C. Highest derivative present

Order = highest derivative. dy/dx = x is 1st order. d²y/dx² + y = 0 is 2nd order. Order determines how many constants in general solution.

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General solution of dy/dx = k y:

  1. y = kx + C
  2. y = C × e^(kx) (exponential)
  3. y = x²
  4. y = k
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Correct answer: B. y = C × e^(kx) (exponential)

We have dy/y = k dx → ln|y|. That is dy/y = kx + C → y. So dy/y = A e^(kx). Famous equation of exponential growth (k > 0) or decay (k < 0). Models population, radioactivity, capacitor discharge.

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Variable separable form:

  1. y' = y
  2. f(x) dx + g(y) dy = 0 (separable)
  3. y² = x
  4. Not solvable
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Correct answer: B. f(x) dx + g(y) dy = 0 (separable)

Equation dy/dx = f(x)g(y) is variable separable: dy/g(y) = f(x) dx. Integrate both sides. Most basic DE technique.

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