Problems

01

Homogeneous; linear quotients

• (a) Solve the homogeneous equation $(x^3+y^3)dx-x{y}^{2}dy=0$.
• (b) Solve $(2x+3y-1)dx-4(x+1)dy=0$ using linear quotients.

02

Reduction

The differential equation $y^{\prime \prime }=1+(y^{\prime }{)}^2$ is missing both $x$ and $y$. It can therefore be solved by reduction in $two$ ways. Solve it using both methods, and check that your solutions are equivalent to each other.

03

Swapping roles

Reverse the roles of independent and dependent variables $x$ and $y$ to solve:

• (a) $y-x{y}^{\prime }=y^{\prime }{y}^2{e}^y$
• (b) $x{y}^{\prime }+2=x^3(y-1)y^{\prime }$