Let Y1 and Y2 be two solutions of a2(x)y” + a1(x)y' + a0(x)y...

let $Y_1$ and $Y_2$ be two solutions of $a_2(x)y'' + a_1(x)y' + a_0(x)y = 0$ (Normal) (a) if $W(y_1, y_2)$ is the Wronskian of $y_1$ and $y_2$, show that $a_2(x)frac{dW}{dx} + a_1(x)W = 0$. (b) show that $W = ce^{-intfrac{a_1(x)}{a_2(x)}dx}$

Transcript

00:01 In this problem we are given the differential equation a2 of x y double dash plus a1 of x y dash plus a0 of x y.

00:12 Y is equal to 0 and that y1 and y2 are two solutions of this differential equation.

00:19 In the first question we are given that w of y1, y2 denotes the ronsky and of the solutions y1 and y2 and then we are asked to prove that a2 of x times d, w .w by by d x plus a1 of x w is equal to 0.

00:39 Now the ron scan of two solutions of the differential equation is determined as the determinant of y1 of x which is the first solution, y2 of x which is the second solution then y1 dash of x which is the derivative of the first solution, y2 dash of x which is a derivative of the second solution.

01:02 Therefore the ron scan is y1 of x times y2 dash of x minus y2 of x times y1 dash of x now take the derivative of this function with respect to x now using product rule of differentiation this is the first function y1 of x times the derivative of y 2 dash of x which is y 2 double dash of x plus the second function y2 dash of x times the derivative of the first function that is y1 dash of x minus again applying product rule this is y2 of x times y 1 double dash of x minus y 2 dash of x times y 1 dash of x which simplifies to y 1 of x times y 2 double dash of x minus y 2 of x times y 1 double dash of x now, since y1 and y2 are solutions of this differential equation, it satisfies this differential equation.

02:06 Now, the differential equation a2 of x, y double dash, plus a1 of x, y dash, plus a0 of x, y is equal to 0 can be also written as y double dash, plus a1 of x divided by a2 of x, y dash, plus a note of x divided by a2 of x, y dash, plus a note of x divided by a2 of x, y is equal to 0.

02:30 Now since y 1 is a solution of this differential equation we have y 1 dash is equal to negative a 1 of x divided by a 2 of x y 1 dash minus a note of x divided by a 2 of x times y 1 similarly y 2 double dash is also negative a 1 of x divided by a 2 of x times y 2 dash minus a note of x divided by a 2 of x times y 2 since both y 1 and y 2 satisfies this differential equation...

Question

let $Y_1$ and $Y_2$ be two solutions of $a_2(x)y'' + a_1(x)y' + a_0(x)y = 0$ (Normal) (a) if $W(y_1, y_2)$ is the Wronskian of $y_1$ and $y_2$, show that $a_2(x)frac{dW}{dx} + a_1(x)W = 0$. (b) show that $W = ce^{-intfrac{a_1(x)}{a_2(x)}dx}$

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