Q$8$: Find the continuous solution of the initial value problem $y^{\text{'}}-2y=g(t),y(0)=-\frac{1}{2},$ where $g(t)=\{\begin{array}{ccc}1& if& 0t<mn>1</mn>\\ e^{2t}& if& 1t\end{array}.$

Q$9$: Find the general solution of the differential equation $x{y}^{\text{'}}=e^{-3x}{y}^2-y+3x{e}^{3x}$ given that $y_1(x)=e^{3x}$ is a solution.

Q$10$: Find the general solution and a singular solution of the given differential equation $y=x{y}^{\text{'}}-2(y^{\text{'}}{)}^2$

Q$11$: Find the general solution of the differential equation $y^{\text{'}}=-2+\frac{x}{sin(2x+y)},($hint: $u=2x+y)$

Q$12$: Find the general solution of the differential equation $2xy+(2y+x^2)y^{\text{'}}=0$

Q$13$: Find the general solution of the differential equation $y^{\text{'}}=\frac{x{e}^x-2y}{x}.$

Q$14$: Find the general solution of the differential equation $(y^5-2xy)dx+(3x^2+x{y}^4)dy=0.$

Q$15$: Use the existence$-$uniqueness theorem to determine $($without solving the problem$)$ an interval in which the solution of the given initial value problem is certain to exist.

a$)(t-3)y^{\text{'}}+ln(t+2)y=2t,y(1)=2$

b$)(t-3)y^{\text{'}}+ln(t+2)y=2t,y(5)=2$