Q1. State whether the following statement are true or false. Justify your answer with the help of a short proof or a counter-example.
- (i)) The initial value problem dy/dx = x² + y², y(0) = 0 has a unique solution in some interval of the form - h<x<h. (75 words)
- (ii)) The orthogonal trajectories of all the parabolas with vertices at the origin and foci on the x-axis is x² + 2y² = c2. (75 words)
- (iii)) The normal form of the differential equation y'' - 4xy' + (4x² -1) y = -3ex² sin 2x is d²v/dx² + v =-3sin 2x, where v=ye⁻x². (75 words)
- (iv)) The solution of the pde ∂z/∂x + ∂z/∂y =z² is z = −[y + f (x - y)]. (75 words)
- (v)) The pde uxx + x² uxy - (x²/4 + 2) uyy = 0 is hyperbolic in the entire xy-plane. (75 words)
- Picard's theorem guarantees unique ODE solution if f and ∂f/∂y are continuous.
- Orthogonal trajectories are found by replacing dy/dx with -dx/dy.
- Normal form v''+Iv=S is derived by substituting y=ve^(∫-P/2 dx) into y''+Py'+Qy=R.
- Method of characteristics solves P ∂z/∂x + Q ∂z/∂y = R using integral curves.
Answer: The following statements regarding differential equations are analyzed for their truthfulness, with justifications provided through proofs or counter-examples, adhering to the principles outlined in the MTE-08 course material.