Differential Equation Initial Value Problem

Question

Here is a pretty standard initial value problem that I'm having a little trouble with. $$(\ln(y))^2\frac{\mathrm{d}y}{\mathrm{d}x}=x^2y$$ Given $y(1)=e^2$, find the constant $C$.

So I separated and integrated to get $\frac{(\ln(y))^3}{3}=\frac{x^3}{3}+C$. Multiplying $3$ to both sides yields $(ln(y))^3=x^3+C$. Here's where I struggled a little. So I take the cube root of both sides to get $\ln y=\sqrt[3]{x^3+C}$ then raised $e$ to both sides to get $y=e^\sqrt[3]{x^3+C}$...? Basically, the $C$ and the $e$ are giving me trouble. I appreciate any pointers.

Answer 1

It may be easier to solve for $C$ at the step

$$(\ln y)^3=x^3+C$$

$$2^3=1^3+C$$.

Everything else looks good.

Answer 2

First of all, the $C$ in your first equation is not the $C$ in your second equation. (If the first $C$ was called $C_1$, the second $C$ would be equal to $3C_1$.)

You have $y(1)=e^2$. Plugging $x=1$ and $y=e^2$ into your last equation yields $e^2=e^\sqrt[3]{1^2+C}$, thus, $2=\sqrt[3]{1+C}$. Can you do the rest?

Answer 3

$$y'(x)\ln^2(y(x))=y(x)x^2\Longleftrightarrow$$ $$\frac{y'(x)\ln^2(y(x))}{y(x)}=x^2\Longleftrightarrow$$ $$\int\frac{y'(x)\ln^2(y(x))}{y(x)}\space\text{d}x=\int x^2\space\text{d}x\Longleftrightarrow$$

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For the LHS, subtitute $u=\ln(y(x))$ and $\text{d}u=\frac{y'(x)}{y(x)}\space\text{d}x$.

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$$\frac{\ln^3(y(x))}{3}=\frac{x^3}{3}+\text{C}$$

Now, use $y(1)=e^2$:

$$\frac{\ln^3(e^2)}{3}=\frac{1^3}{3}+\text{C}\Longleftrightarrow\text{C}=\frac{7}{3}$$

So, we get:

$$\ln^3(y(x))=7+x^3$$

Now, because of the initial condition, you can only use the real solution:

$$y(x)=\exp\left[\sqrt[3]{7+x^3}\right]$$