Problems

Solve IVPs by variable separation methods.

$1\mathrm{.}1$

$(2y-2)\frac{dy}{dx}=3x^{2}4x2,y(1)=-2$

$1\mathrm{.}2$

$\frac{dx}{dt}=4(x^{2}1),x(\frac{}{4})=1$

$1\mathrm{.}3$

$\frac{dy}{dx}=\frac{y^2-1}{x^2-1},y(2)=2$

$1\mathrm{.}4$

$\frac{dy}{dt}2y=1,y(0)=\frac{5}{2}$Find the explicit solution of the initial$-$value problems:

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems $39-42$ find an explicit solution of the given initial$-$value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of $(0,1).$

$39.\frac{dy}{dx}=(y-1{)}^2,y(0)=1$

$40.\frac{dy}{dx}=(y-1{)}^2,y(0)=1\mathrm{.}01$

$41.\frac{dy}{dx}=(y-1{)}^{2}0\mathrm{.}01,y(0)=1$

$42.\frac{dy}{dx}=(y-1{)}^2-0\mathrm{.}01,y(0)=1$

$31.\frac{dy}{dx}=\frac{2x1}{2y},y(-2)=-1$Solve by the variable separation method the ODEs

a$.\frac{dy}{dx}=x\sqrt[\phantom{2}]{1-y^2}$

b$.$ Solve the differential equation

$y^{\text{'}}=x^{2}y.$

$-$Solve the initial value problem $($IVP$).$ Show steps in calculation:

c$.$

$y^{\text{'}}=(xy-2{)}^2,ify(0)=2$Find the explicit solution of the initial$-$value problems:

$4\mathrm{.}1\frac{dy}{dx}=\frac{2x1}{2y},$ for $y(-2)=-1$

$4\mathrm{.}2sin(x)dxydy=0,$ for $y(0)=1$

$334.sinxdxydy=0,y(0)=1$