OpenStudy (unklerhaukus):

Q: Solve the Second Order DE:\[y\frac{\text d^2y}{\text dx^2}+1=\left(\frac{\text dy}{\text dx}\right)^2\] A:\[\pm c\ln(y+\sqrt{y^2+c^2})=x+d\]

OpenStudy (unklerhaukus):

\[\text {let } \frac{\text dy}{\text dx}=p\]\[\frac{\text d^2y}{\text dx^2}=\frac{\text dp}{\text dx}=\frac{\text dp}{\text dy}\frac{\text dy}{\text dx}=\frac{\text dp}{\text dy}p\]\[yp \frac{\text dp}{\text dy}+1=p^2\]\[yp \frac{\text dp}{\text dy}=p^2-1\]\[\frac{p\text dp}{p^2-1}=\frac{\text dy}{y}\]

OpenStudy (unklerhaukus):

\[\frac 12 \int\frac{2p}{p^2-1}\cdot \text dp=\int\frac{\text dy}{y}\]\[\frac 12\ln|p^2-1|=\ln|cy|\]\[p^2-1=c^2y^2\]\[p=\pm\sqrt{c^2y^2+1}\]\[\frac{\text dy}{\text dx}=\pm\sqrt{c^2y^2+1}\]\[\pm\int\frac{\text dy}{\sqrt{c^2y^2+1}}=\int{\text dx}\]

OpenStudy (unklerhaukus):

\[\text{let }y=\frac 1c\tan\theta\qquad\qquad \theta=\arctan(cy)\]\[\text dy=\frac 1c\sec^2\theta\cdot\text d\theta\]\[\pm\int\frac{\frac 1c\sec^2\theta\cdot\text d\theta}{\sqrt{c^2\left(\frac 1c\tan\theta\right)^2+1}}=\int{\text dx}\]\[\pm\frac 1c\int\frac{\sec^2\theta\cdot\text d\theta}{\sqrt{\tan^2\theta+1}}=\int{\text dx}\]\[\pm\frac 1c\int\sec\theta\cdot\text d\theta=\int{\text dx}\]\[\pm\frac 1c\int \sec \theta \times\frac{\sec \theta+\tan \theta}{\sec \theta+\tan \theta}\text d\theta=\int{\text dx}\]\[\pm\frac 1c\int \frac{\sec^2 \theta+\sec \theta\tan \theta}{\sec \theta+\tan \theta}\text d\theta=\int{\text dx}\]

OpenStudy (unklerhaukus):

\[\text{let } u={\sec \theta+\tan \theta}\]\[\text du=\left(\tan \theta\sec x+\sec^2\theta\right)\text d\theta\]\[\pm\frac 1c\int\frac{\text du}{u}=\int{\text dx}\]\[\pm\frac 1c\ln u=x+d\]

OpenStudy (unklerhaukus):

\[\pm\frac 1c\ln|\sec \theta+\tan \theta|=x+d\]\[\pm\frac 1c\ln\left|\sqrt{ 1+c^2y^2}+cy\right|=x+d\]

OpenStudy (unklerhaukus):

\[\pm c\ln\left|\frac 1c\left(\sqrt{ \frac 1{c^2}+y^2}+y\right)\right|=x+d\]\[\pm c\ln\left|\left(\sqrt{ \frac 1{c^2}+y^2}+y\right)\right|-c\ln c=x+d\]\[\pm c\ln\left|\left(\sqrt{ \frac 1{c^2}+y^2}+y\right)\right|=x+d+c\ln c\]

OpenStudy (unklerhaukus):

having bit of trouble with the constants

OpenStudy (zepp):

That looks a bit scary :#