Generated by the Physics Derivation Graph. Eq. \ref{eq:7572748} is an initial equation. $$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x) \label{eq:7572748}$$ Judicious choice as a guessed solution to Eq. \ref{eq:7572748} is $$\psi(x) = a \sin(kx) + b \cos(kx) \label{eq:7572118}$$ A boundary condition for Eq. \ref{eq:7572748} is Eq. \ref{eq:8577781} $$\psi(x=0) = 0 \label{eq:8577781}$$ A boundary condition for Eq. \ref{eq:7572748} is Eq. \ref{eq:8585727} $$\psi(x=W) = 0 \label{eq:8585727}$$ LHS of Eq. \ref{eq:8577781} is equal to LHS of Eq. \ref{eq:7572118}; yields Eq. \ref{eq:7547581}. $$0 = a \sin(0) + b\cos(0) \label{eq:7547581}$$ Simplify Eq. \ref{eq:7547581}; yields Eq. \ref{eq:7572859}. $$0 = b \label{eq:7572859}$$ Substitute RHS of Eq. \ref{eq:7572859} into Eq. \ref{eq:7572118}; yields Eq. \ref{eq:7562671}. $$\psi(x) = a \sin(k x) \label{eq:7562671}$$ LHS of Eq. \ref{eq:8585727} is equal to LHS of Eq. \ref{eq:7562671}; yields Eq. \ref{eq:8577672}. $$0 = a \sin(k W) \label{eq:8577672}$$ Eq. \ref{eq:8577711} is an identity. $$0 = a \sin(n \pi) \label{eq:8577711}$$ Eq. \ref{eq:8577672} is valid when Eq. \ref{eq:8577711} occurs; yields Eq. \ref{eq:9847600}. $$k W = n \pi \label{eq:9847600}$$ Divide both sides of Eq. \ref{eq:9847600} by $$W$$; yields Eq. \ref{eq:9495882}. $$k = \frac{n \pi}{W} \label{eq:9495882}$$ Substitute RHS of Eq. \ref{eq:9495882} into Eq. \ref{eq:7562671}; yields Eq. \ref{eq:3452131}. $$\psi(x) = a \sin(\frac{n \pi}{W} x) \label{eq:3452131}$$ Normalization condition is Eq. \ref{eq:7575626}. $$\int |\psi(x)|^2 dx = 1 \label{eq:7575626}$$ Conjugate $$\psi$$ in Eq. \ref{eq:3452131}; yields Eq. \ref{eq:3452132}. $$\psi(x)^* = a \sin(\frac{n \pi}{W} x) \label{eq:3452132}$$ Swap LHS of Eq. \ref{eq:7575626} with RHS; yields Eq. \ref{eq:5577567}. $$1 = \int |\psi(x)|^2 dx \label{eq:5577567}$$ Expand $$\psi(x)$$ in Eq. \ref{eq:5577567} with conjugate; yields Eq. \ref{eq:0595847}. $$1 = \int \psi(x)\psi(x)^* dx \label{eq:0595847}$$ Substitute LHS of Eq. \ref{eq:3452131} into Eq. \ref{eq:0595847}; yields Eq. \ref{eq:0495950}. $$1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx \label{eq:0495950}$$ Substitute LHS of Eq. \ref{eq:3452132} into Eq. \ref{eq:0495950}; yields Eq. \ref{eq:8478550}. $$1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx \label{eq:8478550}$$ Eq. \ref{eq:1231131} is an identity. $$(\sin(x))^2 = \frac{1 - \cos(2 x)}{2} \label{eq:1231131}$$ Change variable $$\frac{n \pi}{W}x$$ to $$x$$ in Eq. \ref{eq:1231131}; yields Eq. \ref{eq:0100404}. $$\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} \label{eq:0100404}$$ Substitute RHS of Eq. \ref{eq:0100404} into Eq. \ref{eq:8478550}; yields Eq. \ref{eq:9485800}. $$1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx \label{eq:9485800}$$ Divide both sides of Eq. \ref{eq:9485800} by $$a^2$$; yields Eq. \ref{eq:0495054}. $$\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx \label{eq:0495054}$$ Expand integrand of Eq. \ref{eq:0495054}; yields Eq. \ref{eq:0203020}. $$\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx \label{eq:0203020}$$ Eq. \ref{eq:3992939} is an identity. $$\int \cos(a x) dx = \frac{1}{a}\sin(a x) \label{eq:3992939}$$ Change variable $$\frac{2n\pi}{W}$$ to $$a$$ in Eq. \ref{eq:3992939}; yields Eq. \ref{eq:4948377}. $$\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \label{eq:4948377}$$ Eq. \ref{eq:0021030} is an identity. $$\int a dx = a x \label{eq:0021030}$$ Change variable $$1/2$$ to $$a$$ in Eq. \ref{eq:0021030}; yields Eq. \ref{eq:9339495}. $$\int \frac{1}{2} dx = \frac{1}{2} x \label{eq:9339495}$$ Substitute LHS of Eq. \ref{eq:9339495} into Eq. \ref{eq:0203020}; yields Eq. \ref{eq:8584733}. $$\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx \label{eq:8584733}$$ Substitute RHS of Eq. \ref{eq:4948377} into Eq. \ref{eq:8584733}; yields Eq. \ref{eq:0405049}. $$\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W \label{eq:0405049}$$ Simplify Eq. \ref{eq:0405049}; yields Eq. \ref{eq:9493949}. $$\frac{1}{a^2} = \frac{W}{2} \label{eq:9493949}$$ Multiply both sides of Eq. \ref{eq:9493949} by $$a^2\frac{2}{W}$$; yields Eq. \ref{eq:1029384}. $$\frac{2}{W} = a^2 \label{eq:1029384}$$ Take the square root of both sides of Eq. \ref{eq:1029384}; yields Eq. \ref{eq:9394857} and Eq. \ref{eq:9394858}. $$\sqrt{\frac{2}{W}} = a \label{eq:9394857}$$ $$-\sqrt{\frac{2}{W}} = a \label{eq:9394858}$$ Substitute LHS of Eq. \ref{eq:9394857} into Eq. \ref{eq:3452131}; yields Eq. \ref{eq:8474766}. $$\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) \label{eq:8474766}$$ Substitute LHS of Eq. \ref{eq:9394858} into Eq. \ref{eq:3452131}; yields Eq. \ref{eq:8474765}. $$\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) \label{eq:8474765}$$ Eq. \ref{eq:8474765} is one of the final equations. Substitute RHS of Eq. \ref{eq:7572748} into Eq. \ref{eq:7572118}; yields Eq. \ref{eq:2838288}. $$\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right) \label{eq:2838288}$$ Simplify Eq. \ref{eq:2838288}; yields Eq. \ref{eq:8474762}. $$a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx) \label{eq:8474762}$$ Simplify Eq. \ref{eq:8474762}; yields Eq. \ref{eq:1214762}. $$-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x) \label{eq:1214762}$$ Thus we see that LHS of Eq. \ref{eq:1214762} is equal to RHS.