ODEs

Question: Find Orthogonal trajectories to the family of straight lines y=k(x-1) where k in R Solution : Given family of straight lines \[y=k(x-1) \] (1) To obtain Orthogonal trajectories Step…

Question : Which ode does y=sin(log(x)) satisfy? Solution : Given function \[y=sin\left(log\left(x\right)\right)\] differentiating with respect to x \[\frac{dy}{dx}=\cos{\left(\log{\left(x\right)}\right)}\frac{1}{x}\] \[x\ \frac{dy}{dx}=\cos{\left(\log{\left(x\right)}\right)}\] Again differentiating with respect to x \[\frac{dy}{dx}+x\frac{d^2y}{dx^2}=-sin\left(log\left(x\right)\right)\frac{1}{x}=\frac{y}{x}\] \[x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+y=0\] is the…