4.8.4.1 Example 1
Let , then . This simplifies to
Which is separable. Hence
Hence the implicit solution is
The above method is now compared to using d’Alembert for solving the ode, which results after squaring both sides of the given ode. Squaring the ode gives
Where . This is d’Alembert of the form where and . Taking derivative of (2) w.r.t. gives
Using and the above becomes
Which is separable. Solving for gives
Substituting this back into (2) gives
This is an explicit general solution for the ode . The singular solution is found when in (3) which gives
Eq (2) now becomes
However, and this is the problem with squaring the ode, it can be shown that both solution (4) and (5) do not verify the given . What went wrong? They do verify the ode (with minus sign). This example shows why one must be careful when squaring both sides of an ode and solving the squared version. Therefore It is better to avoid the squaring operation and to try to find a method to solve the original ode in its
original form.