# 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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