Tiêu đề Differential-Equations NCERT-Solutions-Class-12-Maths-Chapter-9.pdf ✅

DIFFERENTIAL EQUATIONS MATHEMATICS STUDY MATERIAL NEERAJ K ANAND PARAM ANAND
A A COMPLETE STUDY PACKAGE FOR COMPETITIVE EXAMS Exercise 9.1 Determine order and degree (if defined) of differential equations given in Exercise 1 to 10: 1. + sin (yccc) = 0 Sol. The given D.E. is + sin yccc = 0 The highest order derivative present in the differential equation is and its order is 4. The given differential equation is not a polynomial equation in derivatives (. . . The term sin yccc is a T-function of derivative yccc). Therefore degree of this D.E. is not defined. Ans. Order 4 and degree not defined. 2. yc + 5y = 0 Sol. The given D.E. is yc + 5y = 0. The highest order derivative present in the D.E. is yc § · ̈ ̧ © 1 and so its order is one. The given D.E. is a polynomial equation in derivatives (yc here) and the highest power raised to highest order derivative yc is one, so its degree is one. Ans. Order 1 and degree 1.
A A COMPLETE STUDY PACKAGE FOR COMPETITIVE EXAMS 3. § · ̈ ̧ © 1 + 3s = 0 Sol. The given D.E. is § · ̈ ̧ © 1 + 3s = 0. The highest order derivative present in the D.E. is and its order is 2. The given D.E is a polynomial equation in derivatives and the highest power raised to highest order derivative is one. Therefore degree of D.E. is 1. Ans. Order 2 and degree 1. 4. § · ̈ ̧ ̈ ̧ © 1 + cos = 0 Sol. The given D.E. is § · ̈ ̧ ̈ ̧ © 1 + cos § · ̈ ̧ © 1 = 0. The highest order derivative present in the differential equation is and its order is 2. The given D.E. is not a polynomial equation in derivatives ( . . . The term cos is a T-function of derivative ). Therefore degree of this D.E. is not defined. Ans. Order 2 and degree not defined. 5. = cos 3x + sin 3x Sol. The given D.E. is = cos 3x + sin 3x. The highest order derivative present in the D.E. is and its order is 2. The given D.E. is a polynomial equation in derivatives and the highest power raised to highest order = § · ̈ ̧ ̈ ̧ © 1 is one, so its degree is 1. Ans. Order 2 and degree 1. Remark. It may be remarked that the terms cos 3x and sin 3x present in the given D.E. are trigonometrical functions (but not T-functions of derivatives). It may be noted that § · ̈ ̧ © 1 is not a polynomial function of derivatives. 6. ( yccc)2 + ( ycc)3 + ( yc)4 + y5 = 0 Sol. The given D.E. is ( yccc) 2 + ( ycc) 3 + ( yc) 4 + y5 = 0. ...(i) The highest order derivative present in the D.E. is yccc and its order is 3.
A A COMPLETE STUDY PACKAGE FOR COMPETITIVE EXAMS The given D.E. is a polynomial equation in derivatives yccc, ycc and yc and the highest power raised to highest order derivative yccc is two, so its degree is 2. Ans. Order 3 and degree 2. 7. yccc + 2ycc + yc = 0 Sol. The given D.E. is yccc + 2ycc + yc = 0. ...(i) The highest order derivative present in the D.E. is yccc and its order is 3. The given D.E. is a polynomial equation in derivatives yccc, ycc and yc and the highest power raised to highest order derivative yccc is one, so its degree is 1. Ans. Order 3 and degree 1. 8. yc + y = ex Sol. The given D.E. is yc + y = ex . ...(i) The highest order derivative present in the D.E. is yc and its order is 1. The given D.E. is a polynomial equation in derivative yc. (It may be noted that ex is an exponential function and not a polynomial function but is not an exponential function of derivatives) and the highest power raised to highest order derivative yc is one, so its degree is 1. Ans. Order 1 and degree 1. 9. ycc + ( yc)2 + 2y = 0 Sol. The given D.E. is ycc + ( yc) 2 + 2y = 0. ...(i) The highest order derivative present in the D.E. is ycc and its order is 2. The given D.E. is a polynomial equation in derivatives ycc and yc and the highest power raised to highest order derivative ycc is one, so its degree is 1. Ans. Order 2 and degree 1. 10. ycc + 2yc + sin y = 0 Sol. The given D.E. is ycc + 2yc + sin y = 0. ...(i) The highest order derivative present in the D.E. is ycc and its order is 2. The given D.E. is a polynomial equation in derivatives ycc and yc. (It may be noted that sin y is not a polynomial function of y, it is a T-function of y but is not a T-function of derivatives) and the highest power raised to highest order derivative ycc is one, so its degree is one. Ans. Order 2 and degree 1. 11. The degree of the differential equation § · ̈ ̧ ̈ ̧ © 1 + § · ̈ ̧ © 1 + sin § · ̈ ̧ © 1 + 1 = 0 is (A) 3 (B) 2 (C) 1 (D) Not defined.