MAT244--2019F > Quiz-3

TUT0601 quiz3

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xuanzhong:
Find the Wronskian of two solutions of the given differential equation without solving the equation.
$$cos(t)y^{\prime\prime}+sin(t)y^\prime-ty=0$$

First, we divide both sides of the equation by cos(t):
$$y^{\prime\prime}+\frac{sin(t)}{cos(t)}y^\prime-\frac{1}{cos(t)}y=0$$

Now the given second-order differential equation has the form:
$$L[y]= y^{\prime\prime}+p(t)y^\prime-\ q(t)y=0$$

Then by Abel’s Theorem: the Wronskian W[y1,y2](t) is given by
$$W[y_1,y_2\ ](t)=cexp(-\int p(t)dt)$$
$$=cexp(-\int\frac{sin(t)}/{cos(t)}dt〗)$$
$$=cexp(\int\frac{1}{cos(t)}\ d(cos(t)〗)$$
$$=ce^{ln|cos(t)|}\$$
$$=ccos(t)$$