(1) For each of the following equations, state the order, degree and whether if is linear or non-linear and homogeneous or nonhomogeneous: (a) \( u_{t}-u_{x x}+1=0 \), (b) \( u_{t}-u_{x x}+x u=0 \) (c) \( u_{t}-u_{x x t}+u u_{x}=0 \), (d) \( u_{t t}-u_{x x}+x^{2}=0 \) (e) \( u_{x}+e^{u} u_{y}=0 \) (2) In each of the following, classify the equation as linear, nonlinear but quasi-linear, or fully non-linear: (a) \( u^{2} u_{x x}+u_{y}=\cos (u) \), (b) \( x^{2} u_{x}+y^{2} u_{y}+u_{x y}=2 x y \) (c) \( (x-y) u_{x}^{2}+u_{x y}=1 \), (d) \( x^{2} u_{y y}-y u_{x x}=\tan (u) \) (e) \( u_{x y}-u_{x}^{2}+u_{y}^{2}-\sin \left(u_{x}\right)=0 \), (f) \( u_{x}+u_{y}{ }^{2}-u_{x x}=4 \) (g) \( \frac{u_{y}}{u_{x}}=x^{2} \)

For equation (1): (a) This equation is of order 1, degree 1, linear, and nonhomogeneous due to the constant term \(1\). (b) Here we have order 1, degree 1, linear, and it’s nonhomogeneous because of the term \(x u\). (c) This one is of order 1, degree 1, non-linear, and nonhomogeneous due to the \(u u_{x}\) term. (d) This equation is of order 2, degree 2, linear, and nonhomogeneous because of the \(x^{2}\) term. (e) This is of order 1, degree 1, non-linear, and homogeneous as there are no additional terms. For equation (2): (a) This is classified as fully non-linear due to the presence of \(u^{2} u_{x x}\). (b) This equation is quasi-linear because it is linear in the highest-order derivatives but has nonlinear coefficients. (c) This equation is non-linear due to the \((x-y) u_{x}^{2}\) term. (d) Here, we find a fully non-linear equation because of the \(\tan(u)\) term. (e) This equation is also fully non-linear due to the presence of squared first derivatives and a sine function of one of the derivatives. (f) This is classified as quasi-linear as it features linear terms and may influence the equation’s structure. (g) This equation is non-linear because it represents a relationship involving \(u_{y}\) over \(u_{x}\), creating a geometrical representation.