Answer to Question #104256 in Differential Equations for Ajay

Question

Classify the following statements as true or false giving reasons for your answers.  i) General solution of the differential equations x^2(d^2y/dx^2)^(3) +y(dy/dx)^(4) +y^4 =0 must  contain four arbitrary constants.
(ii) The set of real (or complex) solutions of equation y'(x) + P(x)y(x)=Q(x) forms a
real (or complex) vector space.
(iii) sin xd^2y/dx^2 +dy/dx +y=0 ]0,π[ is linear homogeneous equation.
iv) The partial differential equation
x^2 ∂^2z/∂x^2 +2xy ∂^2z/∂x∂y +y^2 ∂^2z/∂y^2 –x^m y^n =0
is a reducible homogeneous equation.
(v) The partial differential equation u ∂u/∂x =e^y + sin x, u= u(x,y) is a quasi-linear
p.d.e.

Answer 1

I) False. The order of this equation is 2, hence it should contain 2 arbitraty constants

II) True. For all $\lambda, \mu\in C$ If $y_1$ and $y_2$ are solutions of the equation, $\lambda y_1+\mu y_2$ is also the solution.

III)True. It has the form $A(x)y''+B(x)y'+C(x)y=0$ therefore, it is the linear homogeneous equation of second order.

IV) True.the equation can be written in an operational form $x^2D^2+2xyDD'+y^2D'=(x^2D+y^2D')^2$

V)True. since $\sin x+e^y$ is not linear