(19) (a) Suppsoe that y is a solution of the differential equation ay" + by + cy -0. Let 2(2) - y(r- ro) where ro is an arbitrary real number. Show that a:" + be' +ez 0. (b) Let 1(r) wn(r-ro) and z2(r) = 2(r-zo), where yc Cn + ca2 is the general solution to ay" + by +cy 0. Show that se =1 +ea is also a general solution of ay" + by +cy 0.

Q: Show that the y,(2) = K, (1 – 2x* +*-*-)* * (2x -+ 4 x6 + ... 45 4 + K2 (2x 4 8 + ... 3 15 315 is…

A: we have given yh(x)=k11-2x2+23x4-445x6+...+k22x-43x3+415x5-...is complementary solution of…

Q: The given differential equation is already in standard form, as the coefficient of the highest order…

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Q: If x = a + 0 is a singular point of a second-order linear differential equation, then the…

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Q: 1. Express the general solution of the given differential equation on the interval (0, 0) in terms…

A: Solution: (a) 4x2y''+4xy'+64x2-9y=0 Rewrite it as 4x2y''+4xy'+-9+64x2y=0 Divide both sides by 4…

Q: For the differential equation (x2 – 9)y" + (x - 3)y' + 2y = 0, a regular singular point is: X-9 O…

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Q: Which of the following is the differential equation of the family of cubics defined by cy^2 = (x -…

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Q: In this problem, y = 1/(x2 + c) is a one-parameter family of solutions of the first-order DE y' +…

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Q: For the differential equation (x2 – 16)y" + (x – 4)y' + 2y = 0, the singular %3D points are: O x=4…

A: Let us consider the differential equation y''+P(x) y'+Q(x)y=0 Where P(x) and Q(x) are function of x…

Q: The partial differential equation 5x + 8y- = 7+ e10z is elliptic if .......... O 1. x > 0 and y 0 O…

A: We have to choose the correct answer

Q: 2. The differential equation (4x'y -6xy-2x-3dx + (2x*y – 2.x' kdy = 0 is exact because ƏM ON dy

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Q: Verify that the indicated function y = e(x) is an explicit solution of the given first-order…

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Q: [2] Determine the values of r for which the differential equation 2y"+ 5y'- 3y = 0 has solutions of…

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Q: (4x3 – 3y²)dx + 6xydy = 0 Ans: 2x3 + 3y² = 2x %3D

A: first will convert the given differential equation into linear differential equation of first order…

Q: 2. Obtain the general solution of x³y" + 3x2y' + x (+x) y = 0. Hint: The differential equation is…

A: Given: x3y''+3x2y'+x14+xy=0 To find: Bessel function.

Q: 11.By a suitable change of variables it is sometimes possible to transform another differential…

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Q: 2. (a) Check that the first order differential equation 4x- dx dy - 4y =9(/xy² ) is homogeneous and…

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Q: 1 xdy – y(1+ y cos 2x)dx = 0 ;y(0): Ans: 2x + y sin 2x = 0 2

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Q: 1. Verify that y-ciece of the nonhomogenec " as differential equation 5X + be is the general…

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Q: 6.2 What is a differential equation, y depending on x, whose general solution is the function 21…

A: Explanation of the answer is as follows

Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

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Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

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Q: 8. Suppose y1 and y2 are linearly independent, and the Wronskian of them, W (y1, Y2), is identically…

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Q: Verify that the indicated function y = Q(x) is an explicit solution of the given first-order…

A: Solution : when y=125-x2y'=-125-x22×-2xy'=2x25-x22y'=2xy2thus in terms of x,2xy2=dydx

Q: For the differential equation (x2 – 4)y" + (x – 2)y' + 2y = 0, a regular singular point is: x=2 O…

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Q: In this problem, y = c, cos(4x) + c, sin(4x) is a two-parameter family of solutions of the…

A: The second order differential equation is  Y" + 16 Y = 0  But the  solution of differential equation…

Q: For the differential equation (x2 – 9)y" + (x – 3)y' + 2y = 0, a regular singular point is: O x=3…

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Q: 2. Verify that for constants A, B, C and D the function p(x) = Ae* + Be* + C sinx+ D cos x is a…

A: We have to solve given problem:

Q: Suppose that y, denotes a nontrivial solution of the following second-order differential equation:…

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Q: we have y1, y2, .., Yn with domain R such that for any t E R , W (y1, y2, ... , yn) # 0. is there…

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Q: In this problem, y = 1/(x² + c) is a one-parameter family of solutions of the first- order DE y +…

A: Given:- y(x)=1x2+c is one parameter family of solution of DE y'+2xy2=0Initial condition: y(3)=15To…

Q: Consider the differential equation (E): x²y" – 3xy' + 4y = x²Inx Given that y, = x² and y, = x²Inx…

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Q: Consider the differential equation y" - 2y + 17y= 0; ex cos(4x), ex sin(4x), (-∞o, co). Verify that…

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Q: 4. Show that the coefficients of the given differential equation are homogeneous and solve the given…

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Q: Consider the differential equation (E): x²y"-xy' +y 0. Given that y, x and y, xlnx are solutions of…

A: Given solutions are y1 = x  and  y2 = xln(x)

Q: Find the Particular solution of the differential equation : D2+3D+2y=3sin x , given that x=0, y=-0.9…

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Q: Use an appropriate substitution to solve the equation y′−(5/x)y=y^4/x^10,y′−((5/x)y)=y^4/x^13, and…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: if y=x²sin(3inx) is a solution of a second order linear homogeneous differential equation, then the…

A: Given Differential Equation is 2nd order  Option (5) is Correct

Q: Given that y, = 1, y2 = x2,y3 = e* are solutions of a third order differential equation (E). Then…

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Q: For the differential equation (x – 4)y" + (x– 2)y' + 2y = 0, the singular points are: O None of them…

A: Here we use the basic definitions of singular point

Q: In this problem, y c, cos(4x) + c, sin(4x) Is a two-parameter family of solutions of the…

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Q: Verify that the indicated function y = p(x) is an explicit solution of the given first-order…

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Q: In this problem, y = c, cos(5x) + c, sin(5x) is a two-parameter family of solutions of the…

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Q: For the differential equation 2(x – 1)(x+1)*y" +3xy' + (x + 2)y = 0, the regular singular point is:…

A: Topic = Differential equation

Q: 9 In solving certain problems involving support beams in engineering, the fourth-order differential…

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Q: Suppose that yı (x) =x² denotes a nontrivial solution of a homogeneous linear second-order…

A: We have given that second order homogenous differential equation, x2y" + 2xy' - 6y = 0 But we know…

Q: 8. Suppose y1 and y2 are linearly independent, and the Wronskian of them, W(y1, Y2), is identically…

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Q: Verify that the indicated function y g(x) is an explicit solution of the given first-order…

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Q: Verify that the indicated function y = P(x) is an explicit solution of the given first-order…

A: Given: y'=2xy2 and y=116-x2

Q: In this problem, y = c, cos(3x) + c, sin(3x) is a two-parameter family of solutions of the…

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.