Resolver para " \( x \) ": \[ \frac{a x}{a+b}+2 a b=\frac{b x}{a-b}+a^{2}-b^{2} \]

Solve the equation \( \frac{a x}{a+b}+2 a b=\frac{b x}{a-b}+a^{2}-b^{2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{ax}{a+b}+2ab=\frac{bx}{a-b}+a^{2}-b^{2}\) - step1: Multiply both sides of the equation by LCD: \(\left(\frac{ax}{a+b}+2ab\right)\left(a^{2}-b^{2}\right)=\left(\frac{bx}{a-b}+a^{2}-b^{2}\right)\left(a^{2}-b^{2}\right)\) - step2: Simplify the equation: \(\left(a^{2}-ab\right)x+2a^{3}b-2b^{3}a=\left(ab+b^{2}\right)x+a^{4}-2a^{2}b^{2}+b^{4}\) - step3: Move the expression to the left side: \(\left(a^{2}-ab\right)x-\left(ab+b^{2}\right)x=a^{4}-2a^{2}b^{2}+b^{4}-\left(2a^{3}b-2b^{3}a\right)\) - step4: Add and subtract: \(\left(a^{2}-2ab-b^{2}\right)x=a^{4}-2a^{2}b^{2}+b^{4}-\left(2a^{3}b-2b^{3}a\right)\) - step5: Add and subtract: \(\left(a^{2}-2ab-b^{2}\right)x=a^{4}-2a^{2}b^{2}+b^{4}-2a^{3}b+2b^{3}a\) - step6: Divide both sides: \(\frac{\left(a^{2}-2ab-b^{2}\right)x}{a^{2}-2ab-b^{2}}=\frac{a^{4}-2a^{2}b^{2}+b^{4}-2a^{3}b+2b^{3}a}{a^{2}-2ab-b^{2}}\) - step7: Divide the numbers: \(x=a^{2}-b^{2}\) La solución de la ecuación dada es \( x = a^{2} - b^{2} \).