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Magazine Problem 14
Solve the following equation for \(F(t)\) with the condition that \(F(0)=0\) : $$ F^{\prime}(t)=\sin t+\int_{0}^{t} F(t-\beta) \cos \beta d \beta $$
Problem 15
For \(a>0,\) show that from \(L^{-1}\\{f(s)\\}=F(t)\) it follows that $$ L^{-1}\\{f(a s)\\}=\frac{1}{a} F\left(\frac{t}{a}\right) $$.
Problem 15
Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ x^{\prime \prime}(t)+4 x(t)=t+4 ; x(0)=1, x^{\prime}(0)=0 $$.
Problem 16
Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ x^{\prime \prime}(t)-2 x^{\prime}(t)=6-4 t ; x(0)=2, x^{\prime}(0)=0 $$.
Problem 16
For \(a>0\), show that from \(L^{-1}\\{f(s)\\}=F(t)\) it follows that $$ L^{-1}\\{f(a s+b)\\}=\frac{1}{a} \exp \left(-\frac{b t}{a}\right) F\left(\frac{t}{a}\right) $$.
Problem 17
Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ x^{\prime \prime}(t)+x(t)=4 e^{t} ; x(0)=1, x^{\prime}(0)=3 $$.
Problem 17
Evaluate \(L^{-1}\left\\{\frac{e^{-4 s}}{(s+2)^{3}}\right\\}\).
Problem 18
If \(F(t)\) is to be continuous for \(t \geq 0\) and $$F(t)=L^{-1}\left\\{\frac{e^{-3 s}}{(s+1)^{3}}\right\\}$$ evaluate \(F(2), F(5), F(7)\).
Problem 18
Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ x^{\prime \prime}(t)+x^{\prime}(t)-2 x(t)=6 ; x(0)=1, x^{\prime}(0)=1 $$.
Problem 19
Solve the problem by the Laplace transform method. Verify that your solution satisfies the differential equation and the initial conditions. $$ y^{\prime \prime}(x)+9 y(x)=40 e^{x} ; y(0)=5, y^{\prime}(0)=-2 $$.
