Final Answer
Step-by-step Solution
Specify the solving method
Group the terms of the equation
Multiplying the fraction by $-1$
Divide fractions $\frac{1}{\frac{-1}{\left(y-1\right)^2}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Divide fractions $\frac{1}{\frac{1}{\sqrt{x^2+4}}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$
The trinomial $\left(y^2-2y+1\right)$ is a perfect square trinomial, because it's discriminant is equal to zero
Using the perfect square trinomial formula
Factoring the perfect square trinomial
We can solve the integral $\int-\left(y-1\right)^{2}dy$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $y-1$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part
Now, in order to rewrite $dy$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above
Substituting $u$ and $dy$ in the integral and simplify
The integral of a function times a constant ($-1$) is equal to the constant times the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$
Replace $u$ with the value that we assigned to it in the beginning: $y-1$
Solve the integral $\int-\left(y^2-2y+1\right)dy$ and replace the result in the differential equation
We can solve the integral $\int\sqrt{x^2+4}dx$ by applying integration method of trigonometric substitution using the substitution
Now, in order to rewrite $d\theta$ in terms of $dx$, we need to find the derivative of $x$. We need to calculate $dx$, we can do that by deriving the equation above
Substituting in the original integral, we get
Factor the polynomial $4\tan\left(\theta \right)^2+4$ by it's greatest common factor (GCF): $4$
The power of a product is equal to the product of it's factors raised to the same power
Applying the trigonometric identity: $1+\tan\left(\theta \right)^2 = \sec\left(\theta \right)^2$
The integral of a function times a constant ($4$) is equal to the constant times the integral of the function
Simplify $\sqrt{\sec\left(\theta \right)^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$
Multiply $2$ times $\frac{1}{2}$
When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)\sec\left(\theta \right)^2$
Rewrite $\sec\left(\theta \right)^{3}$ as the product of two secants
When multiplying exponents with same base you can add the exponents: $\sec\left(\theta \right)^2\sec\left(\theta \right)$
Solve the integral $\int\sqrt{x^2+4}dx$ and replace the result in the differential equation
Simplify the integral $\int\sec\left(\theta \right)^{3}d\theta$ applying the reduction formula, $\displaystyle\int\sec(x)^{n}dx=\frac{\sin(x)\sec(x)^{n-1}}{n-1}+\frac{n-2}{n-1}\int\sec(x)^{n-2}dx$
Simplify the expression inside the integral
The integral of the secant function is given by the following formula, $\displaystyle\int\sec(x)dx=\ln\left|\sec(x)+\tan(x)\right|$
Apply the trigonometric identity: $\sin\left(\theta \right)\sec\left(\theta \right)^n$$=\tan\left(\theta \right)\sec\left(\theta \right)^{\left(n-1\right)}$, where $x=\theta $ and $n=2$
Express the variable $\theta$ in terms of the original variable $x$
The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
Combine and simplify all terms in the same fraction with common denominator $2$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Solve the integral $4\int\sec\left(\theta \right)^{3}d\theta$ and replace the result in the differential equation