Answer:
Hence, the particular solution of the differential equation is [tex]y = \frac{1}{6} \cdot x^{3} - \frac{3}{4}\cdot x^{2} - x[/tex].
Step-by-step explanation:
This differential equation has separable variable and can be solved by integration. First derivative is now obtained:
[tex]f'' = x - \frac{3}{2}[/tex]
[tex]f' = \int {\left(x-\frac{3}{2}\right) } \, dx[/tex]
[tex]f' = \int {x} \, dx -\frac{3}{2}\int \, dx[/tex]
[tex]f' = \frac{1}{2}\cdot x^{2} - \frac{3}{2}\cdot x + C[/tex], where C is the integration constant.
The integration constant can be found by using the initial condition for the first derivative ([tex]f'(4) = 1[/tex]):
[tex]1 = \frac{1}{2}\cdot 4^{2} - \frac{3}{2}\cdot (4) + C[/tex]
[tex]C = 1 - \frac{1}{2}\cdot 4^{2} + \frac{3}{2}\cdot (4)[/tex]
[tex]C = -1[/tex]
The first derivative is [tex]y' = \frac{1}{2}\cdot x^{2}- \frac{3}{2}\cdot x - 1[/tex], and the particular solution is found by integrating one more time and using the initial condition ([tex]f(0) = 0[/tex]):
[tex]y = \int {\left(\frac{1}{2}\cdot x^{2}-\frac{3}{2}\cdot x -1 \right)} \, dx[/tex]
[tex]y = \frac{1}{2}\int {x^{2}} \, dx - \frac{3}{2}\int {x} \, dx - \int \, dx[/tex]
[tex]y = \frac{1}{6} \cdot x^{3} - \frac{3}{4}\cdot x^{2} - x + C[/tex]
[tex]C = 0 - \frac{1}{6}\cdot 0^{3} + \frac{3}{4}\cdot 0^{2} + 0[/tex]
[tex]C = 0[/tex]
Hence, the particular solution of the differential equation is [tex]y = \frac{1}{6} \cdot x^{3} - \frac{3}{4}\cdot x^{2} - x[/tex].
