Consider the differential equation x2y′′−7xy′+15y=0;x3,x5,(0,∞) Verify that the given functions form a fundamental set o

[answerhappygod]

Consider The Differential Equation X2y 7xy 15y 0 X3 X5 0 Verify That The Given Functions Form A Fundamental Set O 1 (45.61 KiB) Viewed 36 times

Consider the differential equation x2y′′−7xy′+15y=0;x3,x5,(0,∞) Verify that the given functions form a fundamental set of solutions of the differential equation on indicated interval. Form the general solution. Step 1 We are given the following homogenous differential equation and pair of solutions on the given interval. x2y′′−7xy′+15y=0;x3,x5,(0,∞) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c1​ and c2​, not both zero, such that c1​x3+c2​x5=0. While this are different powers of x, we have a formal test to verify the linear independence. Recall the definition of the Wronskian for the case of two functions f1​ and f2′​ each of which have a first derivative. W(f1​,f2​)=∣∣​f1​f1′​​f2​f2′​​∣∣​ By Theorem 4.1.3, if W(f1​,f2​)=0 for every x in the interval of the solution, then solutions are linearly independent. Let f1​(x)=x3 and f2​(x)=x5. Complete the Wronskian for these functions. W(x3,x5)=∣∣​x33x2​x55x4​∣∣​
Find the determinant. W(x5,x3)​=∣∣​x33x2​x55x4​∣∣​=(x3)(5x4)−(3x2)(x5)​ = The Wronskian equal to