How do I appropriately answer Problem #3 from my Differential Equations Textbook? This specific problem is from a section titled, "Series Solutions Near an Ordinary Point, Part II. I'm guessing that what I would need to do is use some type of Power Series that makes me get a's, and then try to determine the initial values. Although, I'm not quite sure. It wouldn't surprise me if that were the case because I have been doing such ways for a while. Except for any type of initial values. series converges for all x.ProblemsIn each of Problems 1 through 3, determine o"(xo), Q"(xo), and0 (x0) for the given point xo if y = 0(x) is a solution of the giveninitial-value problem.b. Show that if a is a nonnegative integer n, thea polynomial solution of degree n. These polynomproperly normalized, are called the Chebyshev poThey are very useful in problems that require aapproximation to a function defined on-1 at Xo. Hence we can continue to differentiate equation (4) indefinitely, determining after eachdifferentiation the successive coefficients as, ds, by setting x equal to Xo-Theorem 5.3.1M is an ondinary point of the differEXAMPLE 1PCetht is, if px)Q00 P0) andof equation (1) isLet yy'(0)o(x) be a solution of the initial value problem (1+x?) y" + 2xy' + 4x2y1. Determine o"(0), o"(0), and o(4)(0).0, y(0)= 0,%3DSolution:where a and a are arbitrary. cThe solutions y, and y form aeach of the seriaof the series for p and q.To find o"(0), simply evaluate the differential equation when x0:(1+0)o"(0) +2.0.6(0) +4 02 0(0) 0,ons yso o"(0) = 0.To find o"(0), differentiate the differential equation with respect to x:To see that y, and y, am()%3D1+b(x-x)Hence y, satisfies the initialconditions yy(x) =0, y(Also note that althougdifferential equation is exceRather, you should substitsthe coefficients so that the(7)(1+x)0"(x) + 2xo"(x) + 2xo"(x) + 20'(x) + 4.x²o'(x) + 8xo(x) = 0.Then evaluate the resulting equation (7) at x=0:%3DO"(0) +20'(0) = 0.%3DThus o"(0) =-20'(0) =-2 (because o'(0) = 1).Finally, to findo(0), first differentiate equation (7) with respect to x:%3Dsection.We will not prove theFuchs What is importanthat the radius of convers(1+x²)6((x) + 2ro"(x) + 4xo"(x) + 40"(x) + (2 + 4x²)o"(x) + 8xo'(x)+ 8xo (x) + 80(x) = 0.%3Dof convergence of the saThis can be donei%3DEvaluating this equation at x =0 we find((0) +66"(0) + 80 (0) = 0.power series for p andconvergence tests forR(x) are polynomialsratio of two polynoma%3D%3D%3DFinally, using 0(0) = 0 and o"(0) = 0, we conclude that o( (0) = 0.tu that we used in determining the a, was that we could

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How do I appropriately answer Problem #3 from my Differential Equations Textbook? This specific problem is from a section titled, &#34;Series Solutions Near an Ordinary Point, Part II. I'm guessing that what I would need to do is use some type of Power Series that makes me get a's, and then try to determine the initial values. Although, I'm not quite sure. It wouldn't surprise me if that were the case because I have been doing such ways for a while. Except for any type of initial values. series converges for all x.ProblemsIn each of Problems 1 through 3, determine o&#34;(xo), Q&#34;(xo), and0 (x0) for the given point xo if y = 0(x) is a solution of the giveninitial-value problem.b. Show that if a is a nonnegative integer n, thea polynomial solution of degree n. These polynomproperly normalized, are called the Chebyshev poThey are very useful in problems that require aapproximation to a function defined on-1 at Xo. Hence we can continue to differentiate equation (4) indefinitely, determining after eachdifferentiation the successive coefficients as, ds, by setting x equal to Xo-Theorem 5.3.1M is an ondinary point of the differEXAMPLE 1PCetht is, if px)Q00 P0) andof equation (1) isLet yy'(0)o(x) be a solution of the initial value problem (1+x?) y&#34; + 2xy' + 4x2y1. Determine o&#34;(0), o&#34;(0), and o(4)(0).0, y(0)= 0,%3DSolution:where a and a are arbitrary. cThe solutions y, and y form aeach of the seriaof the series for p and q.To find o&#34;(0), simply evaluate the differential equation when x0:(1+0)o&#34;(0) +2.0.6(0) +4 02 0(0) 0,ons yso o&#34;(0) = 0.To find o&#34;(0), differentiate the differential equation with respect to x:To see that y, and y, am()%3D1+b(x-x)Hence y, satisfies the initialconditions yy(x) =0, y(Also note that althougdifferential equation is exceRather, you should substitsthe coefficients so that the(7)(1+x)0&#34;(x) + 2xo&#34;(x) + 2xo&#34;(x) + 20'(x) + 4.x²o'(x) + 8xo(x) = 0.Then evaluate the resulting equation (7) at x=0:%3DO&#34;(0) +20'(0) = 0.%3DThus o&#34;(0) =-20'(0) =-2 (because o'(0) = 1).Finally, to findo(0), first differentiate equation (7) with respect to x:%3Dsection.We will not prove theFuchs What is importanthat the radius of convers(1+x²)6((x) + 2ro&#34;(x) + 4xo&#34;(x) + 40&#34;(x) + (2 + 4x²)o&#34;(x) + 8xo'(x)+ 8xo (x) + 80(x) = 0.%3Dof convergence of the saThis can be donei%3DEvaluating this equation at x =0 we find((0) +66&#34;(0) + 80 (0) = 0.power series for p andconvergence tests forR(x) are polynomialsratio of two polynoma%3D%3D%3DFinally, using 0(0) = 0 and o&#34;(0) = 0, we conclude that o( (0) = 0.tu that we used in determining the a, was that we could

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