Answer:
[tex]y(x)=-e^{x-(\frac{11*x^{2} }{2}) }[/tex]
Step-by-step explanation:
This is a separable equation. First divide both sides by y:
[tex]\frac{dy(x)}{\frac{dx}{y(x)} } =-11x+1\\\frac{dy}{y}=(-11x+1)dx[/tex]
Integrate both sides:
[tex]\int\ \frac{dy}{y} \ =\int\ (-11x+1) \, dx[/tex]
[tex]log(y)=-(\frac{11*x}{2} )+x+ c_1[/tex]
Solve for y taking exp to both sides:
[tex]y(x)=c_1*e^{x-(\frac{11*x^{2} }{2}) }[/tex]
Where [tex]c_1[/tex] is an arbitrary constant
Evaluating the initial condition:
[tex]y(0)=c_1*e^{0-(\frac{11*0^{2} }{2}) } =-1[/tex]
[tex]c_1*e^{0} =-1\\c_1*1=-1\\c_1=-1[/tex]
Finally, replacing [tex]c_1[/tex] in the differential equation solution:
[tex]y(x)=-e^{x-(\frac{11*x^{2} }{2}) }[/tex]
