$2$:$12$ PM Mon $17$ Feb

Assignment $1$

Problem $1.$ Find the orders of the following differential equations:

$($i$).$

$7(x-1)\frac{d^{4}y}{d{x}^4}+cosx\frac{dy}{dx}-x{y}^5=12sinx$

$($ii$).$

$65x\frac{d^{5}y}{d{x}^5}+(x+5)(\frac{dy}{dx}{)}^6-9x=3x+24$

Problem $2.$ Prove that $y=\frac{lnx}{x^2}$ is a solution of the following differential equation:

$x^2{y}^{\text{'}\text{'}}+5x{y}^{\text{'}}+4y=0$

Problem $3.$ Solve the following initial value problem:

$\frac{dy}{dx}=x{e}^{-x}$;$y(0)=1$

Problem $4.$ Solve the following separable differential equations:

$(x^2+1)(tany)y^{\text{'}}=x$

Problem $5.$ Find the integrating factor of the following linear differential equation:

$x{y}^{\text{'}}=2y+x^{3}cosx$

Problem $6.$ Solve the following linear differential equation:

$x{y}^{\text{'}}=2y+x^{3}cosx$

Problem $7.$ Show that the function $f(x,y)=\frac{x^{4}y+x^2{y}^3}{xy}$ is homogeneous of degree $3.$

Problem $8.$ Solve the following homogeneous differential equation:

$y{y}^{\text{'}}+x=\sqrt[\phantom{2}]{x^2+y^2}$

Problem $9.$ Prove that the following differential equation is exact:

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

Problem $10.$ Solve the following exact differential equation:

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