Differential equations: midterm

These are exercises for Differential equations: course

1. Solve the differential equation: $y^{\prime}=-6xy.$
2. Solve the following homogeneous equation: $2xy\frac{dy}{dx}=4x^{2}+3y^{2}.$
3. According to Newton's law of cooling, the change of the temperature $T$ of a body immersed in a medium of constant temperature $A$ is described by the differential equation, with respect to time $x$: $\frac{dT}{dx}=k(A-T),k>0.$ Solve it. Interpret your solution to explain why the temperature of the body will become equal to $A,$ eventually.
4. Solve this exact equation $y^{3}dx+3y^{2}xdy=0.$
5. Find all curves perpendicular to the family of curves: $x^{2}y=C.$
6. Solve the differential equation $y^{\prime\prime}+y^{\prime}x=0.$
7. Solve the following differential equation in the complex domain: $y^{\prime\prime}+y^{\prime}+y=0.$
8. Solve the following initial value problem: $y^{\prime\prime}+2y=0,y(0)=0,y^{\prime}(0)=1.$
9. Set up and solve the differential equation that describes the motion of an object of mass $M$ suspended vertically by a spring with Hooke constant $k.$