Initial value problems are problems is an ordinary differential equation with an initial condition which specifies the value of the unknown function at a given point in the domain.

Solved Problems

Problem 1: Find the function $y_1(t)$​​​​ which is the solution of $9y''-48y'+28y=0$​​​​ with initial conditions $y_1(0)=1$​​​ and $y_1'(0)=0$​​​​

Solution 1

• Step 1: Find general solution first. Solve auxilary equation: $9m^2-48m+28=0$​​​ , Solutions are $\frac{14}{3}$​​​ and $\frac{2}{3}$​​​​

• Step 2: Solution will look like $y_1(t) = C_1e^{2t/3}-C_2e^{14t/3}$​​​​

• Step 3: Find $y_1'(t)$: $\frac{2}{3}C_1e^{2t/3} - \frac{14}{3}C_2e^{14t/3}$​

• Step 4: $y_1(0)=1$​​​, so $C_1 - C_2=1$​​​​​ and $y_1'(0)$​, so $\frac{2}{3}C_1 - \frac{14}{3}C_2=0$

• Step 5: Solve system of equations [^1], use any method:

  $$\begin{bmatrix} 1 && -1 \\ \frac{2}{3} && \frac{14}{3}\end{bmatrix}^{-1}\begin{bmatrix} 1 \\ 0\end{bmatrix}$$

  solving this gives $C_1=\frac{7}{6}$​ and $C_2=\frac{1}{6}$

• Answer is $(7/6)e^{2t/3}-(1/6)e^{14t/3}$​​​​