Exercise 9.17. Consider the equation 2² f əxəy (a) Show that the cubic polynomials = 0 for an unknown function f. Unlike

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Exercise 9.17. Consider the equation 2² f əxəy (a) Show that the cubic polynomials = 0 for an unknown function f. Unlike single-variable differential equations (which you probably encountered in high school or in Math 20, and which are studied in greater depth in Math 53), such "partial differential equations" typically have a huge variety of solutions. This exercise illustrates that variety. Copyright © 2021 Stanford Math Dept. This content is protected and may not be shared, uploaded, or distributed. Page 182 f(x, y) = A + Bx + Cy+Dx² + Exy + Fy² +Gx³ + Hr²y + Ixy² + Jy³ (with constants A, B,..., I, J) that satisfy this partial differential equation are those with E = H = I= 0 (i.e., the ones with no "mixed" terms "y" with n, m > 0). Hint: plug this expression into the desired equation and see what conditions emerge on the coefficients. (b) Show that any function of the form f(x, y) = g(x) + h(y), where g and h are functions of one variable (differentiable as often as we want), is a solution of this partial differential equation.