Exercises

01

Homogeneous equations

For the following differential equations, show that they are homogeneous, and then solve them by substitution and separation:

• (a) $(x^2-2y^2)dx+xydy=0$
• (b) $x^2{y}^{\prime }=3(x^2+y^2)\cdot {\tan }^{-1}(y/x)+xy$
• (c) $x{y}^{\prime }=\sqrt{x^2+y^2}$

02

Method of integrating factors

Show that if the function $g(x)=\frac{1}{N}(M_y-N_x)$ depends on $x$ alone, we can use $\mu (x)=e^{\int g(x)dx}$ as an integrating factor and the resulting differential equation becomes exact.

03

Integrating factor: converting to exact

Convert the following to exact equations by finding and applying integrating factors.

• (a) $(x^4-x+y)dx-xdy=0$
• (b) $(2y^2+2y+4x^2)dx+(2xy+x)dy=0$

04

Bernoulli substitution

Show that substituting $v=y^{1-n}$ does indeed convert the Bernoulli equation to a linear equation.

05

Solving Bernoulli equations

Solve the Bernoulli equation $ydx+xdy=x{y}^{2}dx$.

06

Swapping roles

Reverse the roles of $x$ and $y$ to solve: $(e^y-2xy)y^{\prime }=y^2$.

07

Homogeneous

Solve the following homogeneous equations by substitution of $z=y/x$. You should first write the equation using a single quotient and then divide above and below by $x$.

• (a) $x\sin (y/x)\frac{dy}{dx}=y\sin (y/x)+x$
• (b) $x{y}^{\prime }=2x-6y$

08

Integrating factors, conversion to exact

Solve the following differential equations by finding an integrating factor $\mu (x)$ or $\mu (y)$.

• (a) $-2xydx+(3x^2-y^2)dy=0$
• (b) $(y\ln y-2xy)dx+(x+y)dy=0$

09

Bernoulli

Solve the following Bernoulli equation: $x{y}^2{y}^{\prime }+y^3=x\cos x$.