Cauchy partial differential equation

prove d/dt(u(x(t),t))+tanh(x(t))(d/dx(u(x(t),t)))=0 u(x(t),0)=a(x(t)) limit as x tends to infinity of u(x,t)=0 has at most one solution.Explain why there is no boundary condition at x=0 and find a solution for the special case a(x(t))=sinh(x(t))

This problem was summitted and it turn out incomplete, please demonstrate and solve this problem in detail

The meteorologists at the National Interagency Fire Center had pizza delivered to their operations center. Their lunch consisted of pizza, milk and gelatin. One slice of cheese pizza contains 290 calories, 15g of protein, 9g of fat, and 39g of carbohydrates. One-half cup of gelatin dessert contains 70 calories, 2g of protein, 0g of fat, and 17g of carbohydrates. One cup of whole milk contains 150 calories, 8g of protein, 8g of fat, and 11g of carbohydrates. a) Write 1 x 4 matrices P, G, and M that represent the nutritional values of each food. b) Find 3P+2G+2M and tell what the entries represent.

Heat Equation

Please see the attached file.

Seperable Equations

Please help solve this seperable equation and a possible suggestion on how to look for factors to multiply by in order to make it exact. 8*cos^2(y)dx+csc^2(x)dy=0

Solving differential equation

y'=ty^3 with y(0)= -1 over [-3/4, 3/4] Having trouble solving for y