Solve the differential equation dx/dy = a * x^(2/3), where a is a constant.

Step-by-step solution

1. 1

   Separate the variables. Dividing both sides by $x^{2/3}$ and multiplying by $dy$ puts each variable with its own differential:

   $x^{-2/3}\,dx=a\,dy.$

   Note that dividing by $x^{2/3}$ is only legal when $x\ne0$ — the case $x=0$ has to be examined separately, and it turns out to matter here.

2. 2

   Integrate the left side with the power rule. With $n=-\tfrac23$, the new exponent is $n+1=\tfrac13$:

   $\int x^{-2/3}\,dx=\frac{x^{1/3}}{1/3}=3x^{1/3}.$

   Dividing by $\tfrac13$ means multiplying by $3$ — the source of the factor that later becomes $\tfrac{a}{3}$.

3. 3

   Integrate the right side and combine. Since $a$ is constant, $\int a\,dy=ay$, so

   $3x^{1/3}=ay+C.$

   One arbitrary constant suffices for a first-order equation.

4. 4

   Solve for $x$ explicitly. Dividing by $3$ and absorbing the constant:

   $x^{1/3}=\frac{a}{3}y+C_{1}\;\Longrightarrow\;x=\left(\frac{a}{3}y+C_{1}\right)^{3}.$

   Cubing is safe with no $\pm$ ambiguity, because the cube function is one-to-one on the reals — unlike squaring.

5. 5

   Add the singular solution and verify. The constant function $x\equiv0$ satisfies the equation ($\frac{dx}{dy}=0$ and $a\cdot0^{2/3}=0$) but is not of the general form for any finite $C_{1}$, so it is a genuine singular solution. It also makes this a standard example of non-uniqueness: through the point $x=0$ both $x\equiv0$ and $x=\left(\tfrac{a}{3}y\right)^{3}$ pass, because $x^{2/3}$ is not Lipschitz at $0$. Numeric check with $a=1.7$, $C_{1}=0.4$ at $y=1.1$: $\frac{dx}{dy}=1.7802589$ and $ax^{2/3}=1.7802589$ ✓.