f$)$ Use Inverse differential operator to solve the equation

$(5$ Marks$)$

$(D^2-3)y=x^3+x-5$

g$)$ Using Taylor series expansion, find a power series solution of the initial value problem

$(3$ Marks$)$

$y^{\text{'}\text{'}}-x{y}^{\text{'}}+4y=0,y(0)=1,y^{\text{'}}(0)=-1$

QUESTION TWO $(20$ Marks$)$

a$)$ Use method of separation of vatiable to solve the differential equation

$(x-3)y^{4}dx-x^4(y^2-3)dy=0$

b$)$ Use un determined coefficient method to solve $\frac{d^{2}y}{d{x}^2}+16y=x^3-1$

$(4$ Marks$)$

c$)$ Use method of variation of parameter to solve

$y^{\text{'}\text{'}}+9y=e^x$

$(10$ Marks$)$

QUESTION THREE $(20$ marks$)$

a$)$ If the population of a town doubles in $40$ years in how many years will it triple under the assumptions that the rate of increase is proportional to the number of inhabitants?

$(6$ marks$)$

b$)$ Find the integrating factor of the non$-$exact equation $4x^{3}y+(2x^4-5)\frac{dy}{dx}=0$ Hence solve it

$(6$ marks$)$

c$)$ Use power series solution to find the general solution of $y^{\text{'}\text{'}}+2x{y}^{\text{'}}-3y=0$ near $x=0(8$ marks$)$

QUESTION FOUR $(20$ marks$)$

a$)$ Determine the independent and dependent variables and hence classify the differential equation given below to orders, degree, and linearity.

$y\frac{d^{2}x}{d{y}^2}=x+4y^3$

b$)$ Solve the homogeneous equation

$xy\frac{dy}{dx}+x^2+y^2=0$

$(6$ marks$)$solve all questions I have paid for uyou to do your work