Question

Find the solution of the given initial value problem in explicit form. y' = (1 – 11x)y?, y(0) = -1 Enclose numerators and denominators in parentheses. For example, (a - b)/(1 + n). y(x) = Q

Answer 1

Answer:

$y(x)=-e^{x-(\frac{11*x^{2} }{2}) }$

Step-by-step explanation:

This is a separable equation. First divide both sides by y:

$\frac{dy(x)}{\frac{dx}{y(x)} } =-11x+1\\\frac{dy}{y}=(-11x+1)dx$

Integrate both sides:

$\int\ \frac{dy}{y} \ =\int\ (-11x+1) \, dx$

$log(y)=-(\frac{11*x}{2} )+x+ c_1$

Solve for y taking exp to both sides:

$y(x)=c_1*e^{x-(\frac{11*x^{2} }{2}) }$

Where $c_1$ is an arbitrary constant

Evaluating the initial condition:

$y(0)=c_1*e^{0-(\frac{11*0^{2} }{2}) } =-1$

$c_1*e^{0} =-1\\c_1*1=-1\\c_1=-1$

Finally, replacing $c_1$ in the differential equation solution:

$y(x)=-e^{x-(\frac{11*x^{2} }{2}) }$