fill in the blank. Calc 2 Part a and b Consider the following differential equation by"...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
First, verify that y(x) satisfies the given differential equation. Then, determine a value of the constant C so that y(x) satisfies the given initial condition. Use a computer or graphing calculator to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. y' =y+3; y(x) = CeX-3; y(0) = 8 What step should you take to verify that the function is a solution to the given differential equation? O A. Differentiate...
Consider the differential equation: xạy" + 15xy' + 48y = 0. Find all values of r such that y=r" satisfies the differential equation for x > 0. If there is more than one correct answer, enter your answers as a comma separated list. r=
Verify by substitution that the given function is a solution of the given differential equation. Note that any primes denote derivatives with respect to x. y' = 4x3y = x + 6 What step should you take to verify that the function is a solution to the given differential equation? O O O O A. Substitute the given function into the differential equation, B . Integrate the function and substitute into the differential equation C . Determine the first and...
Fill in the blanks (1 point) A Bernoulli differential equation is one of the form dy + P()y= Q(Cy" (*) dr Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=yl-n transforms the Bernoulli equation into the linear equation du dac + (1 - n)P(x)u= (1 - nQ(2). Consider the initial value problem xy' +y= -6xy?, y(1) = -2. (a) This differential equation can be written in the form...
a,b, and c 1. Determine if the following function is a solution of a differential equation. a) y = 4e2' + Be for 6y"- 7y3y = 0 (3pts) b) y = Ae* + Be*** + 2 sin (3x) for y"+y'-6y=6cos(3x) – 30sin (3x) (3pts) c) y = Ae?" + Bx? for y"- 3y + 2y = 0 (3pts)
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we have a solution to this ODE in the form y-Σχ0CnXntr where cn 0. By considering the first term of this series show that r must satisfy r2-4-0 (and hence that r = 2 or r =-2) (b) Show that any solution of the form y-ca:0G,2n-2 must satisfy C0 (c) From the theory about singular solutions we know that a linearly...
Suppose that the differential equation dy/dx=f(x,y) satisfies all the hypotheses of Theorem 1.2.1, i.e. f and df/dy are continuous on a rectangle R ⊂ R^2 . Explain why two solution curves cannot intersect at a point (x0,y0) R ( x 0 , y 0 ) ∈ R .
17. Consider the differential equation given by dy/dx = xy/2 (A) On the axes provided, sketch a slope field for the given differential equation. (B) Let f be the function that satisfies the given differential equation. Write an equation for the tangent line to the curve y (x) through the point (1, 1). Then use your tangent line equation to estimate the value of f(1.2) (C) Find the particular solution y=f(x) to the differential equation with the initial condition f(1)=1. Use your solution...
both a and b ,thanks 2. i)Suppose that f :R- R is differentiable and P(x,y) is defined bu Fa,y)-(2-3y). a) Show that F satisfies the partial differential equation 230 b) Given that F(r,0)sin(2x) for all z E R, find a formula for F(z,y).