Find the solution of the initial-value problem $\frac { d y } { d x } = x \sin \left( x ^ { 2 } \right)$ , $y ( 0 ) = 0$ .
A) $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) - \frac { 1 } { 2 }$
B) $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right)$
C) $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$
D) $\frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$
E) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) - \frac { 1 } { 2 }$
F) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right)$
G) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$
H) $\frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$

Question

Find the solution of the initial-value problem $\frac { d y } { d x } = x \sin \left( x ^ { 2 } \right)$ , $y ( 0 ) = 0$ .
A) $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) - \frac { 1 } { 2 }$ 
B) $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right)$ 
C) $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$ 
D) $\frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$ 
E) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) - \frac { 1 } { 2 }$ 
F) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right)$ 
G) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$ 
H) $\frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$

Quizplus · Accepted Answer

The solution involves integrating the given differential equation. The integral of $x \sin(x^2)$ with respect to $x$ is $-\frac{1}{2} \cos(x^2)$. Applying the initial condition $y(0) = 0$, we find the constant of integration to be $\frac{1}{2}$. Thus, the solution is $-\frac{1}{2} \cos(x^2) + \frac{1}{2}$.

Find the Solution of the Initial-Value Problem dydx=xsin⁡(x2)\frac { d y } { d x } = x \sin \left( x ^ { 2 } \right)dxdy​=xsin(x2)

Find the Solution of the Initial-Value Problem $\frac { d y } { d x } = x \sin \left( x ^ { 2 } \right)$