Tiêu đề 9. DIFFERENTIAL EQUATIONS.pdf ✅

TOPIC-1 Basics of Differential Equations Concepts Covered:  Order of differential equation  Degree of differential equation Revision Notes Differential Equation: In Mathematics, a differential equation is an equation with one or more derivatives of a function. The derivative of the function is given by dy dx . In other words, it is defined as the equation that contains derivatives of one or more dependent variables with respect to one or more independent variables. Order of a differential equation: It is the order of the highest order derivative appearing in the differential equation. Degree of a differential equation: It is the degree (power) of the highest order derivative, when the differential coefficients are made free from the radicals and the fractions. z We shall prefer to use the following notations for derivatives. z dy dx = y', d y dx 2 2 = y'', d y dx 3 3 = y''' z For derivatives of higher order, it will be in convenient to use so many dashes as super suffix therefore, we use the notation yn for nth order derivative d y dx n n . z Order and degree (if defined) of a differential equation are always positive integers. [APQ 2023-24; SQP 2023-24; Board, 2023] Example-1 Find the order and degree of the differential equation ym + y2 + ey' = 0. Sol. The order of the differential equation ′′′ + + = ′ y y e 2 y 0 is 3, as the highest order of derivative is y′′. The degree of the differential equation is not defined, as the differential equation is not a polynomial. SUBJECTIVE TYPE QUESTIONS Very Short Answer Type Questions (1 mark each) 1. Find the order and the degree of the differential equation x2 d y dx dy dx 2 2 2 4 = +1               . R [Delhi Set-1, 2019] Topper's Answer, 2019 DIFFERENTIAL EQUATIONS LEARNING OUTCOMES After going through this Chapter, the student would be able to understand:  Basics of differential equations  Solution of differential equations by method of separation of variables  Solution of homogeneous differential equation  Solution of linear differential equation of the type: dy dx + = py q, where p and q are functions of x or constants. dy dx + = px q, where p and q are functions of y or constants. 9 CHAPTER LIST OF TOPICS Topic-1: Basics of Differential Equations Topic-2: Variable Separable Methods Topic-3: Linear Differential Equations Topic-4: Homogeneous Differential Equations

Differential Equations (b)Particular Solution: Solution obtained by giving particular values to the arbitrary constants in the general solution of a differential equation is called a particular solution e.g., y = 3 cos x + 2 sin x is a particular solution of the differential equation d y dx y 2 2 + = 0. [SQP 2020-21] (c) Solution of Differential by Variable Separable Method: A variable separable form of the differential equation is the one which can be expressed in the form of f(x)dx = g(y)dy. The solution is given by f x( )dx = + g y( )dy k ∫ ∫ where k is the constant of integration. [APQ Set-1, 2023-24] Example-2 Find the general solution of the differential equation dy dx = x . Sol. Separate the variables of the differential equation. dy = xdx − dy ∫ =− xdx ∫ y = x c 2 2 + KEY-TERM Arbitrary Constants: Arbitrary constant is a symbol that can be assigned different values. The value of the constant is not affected by changes in the values of the equation's variables. SUBJECTIVE TYPE QUESTIONS Very Short Answer Type Questions (1 mark each) 1. Find the solution of the differential equation dy dx = x e3 -2y . R&U [Outside Delhi Set 1, 2, 3 Comptt., 2015] Sol. dy e ∫ −2y = 3 x dx ∫ 2y e dy ∫ = 3 x dx ∫ 1⁄2 or e 2y 2 = x C 4 4 + or 1 2 2 e y = x C 4 4 + 1⁄2 2e 2y = x4 + C1 where (C1 = 4C) [Marking Scheme OD Set-1, 2, 3, 2015] 2. How many arbitrary constants are there in the particular solution of the differential equation dy dx = − = 4 0 xy y = 1 2 ; ( ) R&U [SQP 2020-21] stants in a particular solution of a differential equa- tion is always equal to 0. Concept Applied The number of arbitrary con- 3. Write the solution of the differential equation dy dx y = − 2 . R&U [Foreign 2015] Sol. Given differential equation is dy dx = 2–y On separating the variables, we get 2y dy = dx On integrating both sides, we get 2y dy ∫ = dx ∫ or 2 2 y log = x + C1 or 2y = x log 2 + C1 log 2 \ 2y = x log 2 + C, where C = C1 log 2 1 [Marking Scheme, Foreign, 2015] Short Answer Type Questions-I (2 marks each) 1. Find the general solution of the differential equation: edy/dx = x2 . R&U [Outside Delhi Set-1, 2021-22] Sol. Given differential equation is e dy/dx = x2 Taking log both sides, we get dy dx log e = 2 logx Þ dy dx = 2 logx [ loge = 1] Þ dy = 2 logx dx On integrating both sides, we get dy ∫ = 2 log x dx ∫ Þ y = 2 1.log x dx ∫ 1 Þ y = log ( x dx log ) . d dx 1 1 − x d ( ) x dx      ∫∫∫  [Using integration by parts] Þ y = 2 1 log ( x x) ( ) x − x dx      ∫  Þ y = 2[xlogx – x] + C Þ y = 2x(logx – 1) + C 1 [Marking Scheme 2022-23] 2. Find the general solution of the differential equation: log . dy dx ax by       = + R&U [Outside Delhi Set-2, 2021-22] Scan this Variable Separable Form This Question is for practice and its solution is given at the end of the chapter.