Find all solutions of differential equation

Reginald Metcalf
2022-01-22
Answered

Explicit differential equations

Find all solutions of differential equation

${y}^{\prime 2}-(x+y){y}^{\prime}+xy=0$ ?

Find all solutions of differential equation

You can still ask an expert for help

Ella Williams

Answered 2022-01-22
Author has **28** answers

Your second equation is not ${y}^{\prime}=0$ as you write in the question, but ${y}^{\prime}-y=0$ , in other words ${y}^{\prime}=y$ . This has the well-known solution $y={C}_{2}{e}^{x}$ .

So now you have the solutions$y=\frac{{x}^{2}}{2}+{C}_{1}$ and $y={C}_{2}{e}^{x}$ . Now, for the most difficult part of the trick, you need to find all ways to glue intervals of these solutions together so the derivative matches across the glue point ... which means (consult the differential equations again!) that the gluing point(s) has to lie on the line $x=y$ .

So now you have the solutions

Stella Calderon

Answered 2022-01-23
Author has **35** answers

The second part is $y-y=0\Rightarrow y=y\Rightarrow y={C}_{2}{e}^{x}$

RizerMix

Answered 2022-01-27
Author has **438** answers

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where $y=ax+b$ is a particular solution.

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