JEE Main Integral Calculus & Differential Equations: Differential Equations – Chapter Test

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Difficulty: 10 medium · 10 hard

Differential Equations MCQ questions with answers

The first 3 of 20 questions. Unlock the series for the rest, each with a solution.

1. Q1. The order and degree of the differential equation (d²y/dx²)² + cos(dy/dx) = 0 are respectively:

   अवकल समीकरण (d²y/dx²)² + cos(dy/dx) = 0 की कोटि (order) और घात (degree) क्रमशः हैं:

   • A. order 2, degree 2 / कोटि 2, घात 2
   • B. order 2, degree not defined / कोटि 2, घात परिभाषित नहीं
   • C. order 2, degree 1 / कोटि 2, घात 1
   • D. order 1, degree 2 / कोटि 1, घात 2
   Solution

   The highest order derivative is d²y/dx², so the order is 2. Because the term cos(dy/dx) cannot be expressed as a polynomial in the derivatives, the degree of this equation is not defined.

   उच्चतम कोटि का अवकलज d²y/dx² है, अतः कोटि 2 है। चूँकि पद cos(dy/dx) को अवकलजों के बहुपद के रूप में व्यक्त नहीं किया जा सकता, इसलिए इस समीकरण की घात परिभाषित नहीं है।

2. Q2. Which of the following functions is NOT a solution of the differential equation d²y/dx² + y = 0?

   निम्नलिखित में से कौन-सा फलन अवकल समीकरण d²y/dx² + y = 0 का हल नहीं है?

   • A. y = sin(x + π/4)
   • B. y = x sin x
   • C. y = cos x − 3 sin x
   • D. y = 2 sin x
   Solution

   Every function of the form A sin x + B cos x satisfies the equation, which covers the other three options (sin(x + π/4) expands into such a combination). For y = x sin x, y'' + y = 2cos x ≠ 0, so it is not a solution.

   A sin x + B cos x रूप का प्रत्येक फलन इस समीकरण को संतुष्ट करता है, जिसमें शेष तीनों विकल्प आ जाते हैं (sin(x + π/4) का प्रसार ऐसे ही संयोजन में होता है)। y = x sin x के लिए y'' + y = 2cos x ≠ 0, अतः यह हल नहीं है।

3. Q3. Arrange the following differential equations in increasing order of their order: P. d²y/dx² + 3(dy/dx) = 0; Q. dy/dx = sin x; R. d⁴y/dx⁴ = x; S. (d³y/dx³)² + y = 0. The correct sequence is:

   निम्नलिखित अवकल समीकरणों को उनकी कोटि के बढ़ते क्रम में व्यवस्थित कीजिए: P. d²y/dx² + 3(dy/dx) = 0; Q. dy/dx = sin x; R. d⁴y/dx⁴ = x; S. (d³y/dx³)² + y = 0. सही अनुक्रम है:

   • A. Q, P, S, R
   • B. P, Q, S, R
   • C. Q, P, R, S
   • D. Q, S, P, R
   Solution

   The orders are P: 2, Q: 1, R: 4, S: 3. Arranged by increasing order (1, 2, 3, 4) the sequence is Q, P, S, R.

   कोटियाँ हैं P: 2, Q: 1, R: 4, S: 3। बढ़ती कोटि (1, 2, 3, 4) के अनुसार व्यवस्थित करने पर अनुक्रम Q, P, S, R है।

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