Consider the differential equation

Name Math 264 Summer 2021 Homework Set 2 Show all work! 1. Solve the initial-value problem: dy dx + 3 x y = x2 + x; 1 y (1) = 4 (4 pts.) 2. Consider the equation (x4 + ex + xy 2 ) dx + (x2 y + sin y ) dy = 0. (a) Show that the equation is exact. (b) Solve the di erential equation. (1 pt.) (4 pts.) 2 3. Show that the function f (x; y ) = xy 3 y 4 is homogeneous. (2 pts.) 4. Use an appropriate substitution to solve the di erential equation (4 pts.) 3 dy 2 dx x y = 2 3 x y . 5. Suppose that when a cake is taken out of an oven, its temperature is 340 F. Four minutes later, its temperature is 240 F. If the room temperature is 70 F, how long will it take for the cake to cool to 75 F (to the nearest minute)? (5 pts.) 4 Consider the differential equation [x– (x + y3)23] dx + y2 dy = 0. This is not an exact equation, but we may rearrange the equation as follows. x dx + y2 dy = dx (x3 + 23)2/3 1 that discuss how one could solve the differential equation.
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