Solve frac{d^{2}(x^{2}sin(2x)e^{x})}{2dx^{2}}-d(x^2e^xsin2x)/dx+5/2x^2e^xsin2x

$$\frac { d ^ { 2 } ( x ^ { 2 } \sin ( 2 x ) e ^ { x } ) } { 2 d x ^ { 2 } } - \frac { d ( x ^ { 2 } e ^ { x } \sin 2 x ) } { d x } + \frac { 5 } { 2 } x ^ { 2 } e ^ { x } \sin 2 x$$

Factor

$$e^{x}\left(\frac{3}{2}x^{2}\sin(2x)-2x^{2}\cos(2x)+\frac{1}{2}d\sin(2x)-2x\sin(2x)\right)$$

Solution Steps

Cancel out dx^{2} in both numerator and denominator.

$$factor(\frac{d\sin(2x)e^{x}}{2}-\frac{\mathrm{d}(x^{2}e^{x}\sin(2x))}{\mathrm{d}x}+\frac{5}{2}x^{2}e^{x}\sin(2x))$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \frac{\mathrm{d}(x^{2}e^{x}\sin(2x))}{\mathrm{d}x} times \frac{2}{2}.

$$factor(\frac{d\sin(2x)e^{x}}{2}-\frac{2\frac{\mathrm{d}(x^{2}e^{x}\sin(2x))}{\mathrm{d}x}}{2}+\frac{5}{2}x^{2}e^{x}\sin(2x))$$

Since \frac{d\sin(2x)e^{x}}{2} and \frac{2\frac{\mathrm{d}(x^{2}e^{x}\sin(2x))}{\mathrm{d}x}}{2} have the same denominator, subtract them by subtracting their numerators.

$$factor(\frac{d\sin(2x)e^{x}-2\frac{\mathrm{d}(x^{2}e^{x}\sin(2x))}{\mathrm{d}x}}{2}+\frac{5}{2}x^{2}e^{x}\sin(2x))$$

Factor out \frac{1}{2}.

$$\frac{d\sin(2x)e^{x}-2\frac{\mathrm{d}(x^{2}e^{x}\sin(2x))}{\mathrm{d}x}+5x^{2}e^{x}\sin(2x)}{2}$$

Evaluate

$$\frac{e^{x}\left(4\left(x\sin(x)\right)^{2}-4\left(x\cos(x)\right)^{2}+3x^{2}\sin(2x)+d\sin(2x)-4x\sin(2x)\right)}{2}$$